On linear-quadratic optimal control of implicit difference equations

On linear-quadratic optimal control of implicit difference equations Abstract In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur’e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur’e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context. 1. Introduction In this article we revisit the discrete-time linear-quadratic optimal control problem that is minimizing a quadratic cost functional given by $$ \sum_{j=0}^{\infty} \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right)^{\ast} \left[\begin{array}{@{}cc@{}} Q & S\\ S^{\ast} & R \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right) $$ subject to the implicit difference equation (IDE) \begin{align} E\sigma x_{j} = Ax_{j} + Bu_{j}, \end{align} (1) with the initial condition Ex0 = Ex0 and the stabilization condition $$\lim _{j \to \infty } Ex_{j} = 0$$. Here $$\sigma $$ denotes the shift operator, i.e. $$\sigma x_{j} = x_{j+1}$$. Moreover, $$\left (x_{j}\right )_{j}\in (\mathbb{K}^{n})^{\mathbb{N}_{0}}$$ is the state sequence, and $$\left (u_{j}\right )_{j}\in (\mathbb{K}^{m})^{\mathbb{N}_{0}}$$ is the input sequence. this work we will further assume that the matrix pencil $$zE-A\in \mathbb{K}[z]^{n\times n}$$ is regular, i.e. $${\det (zE-A)\not \equiv 0}$$. Such discrete-time systems often appear during the time-discretization (Brenan et al., 1996) or discrete-time lifting (Kahane et al., 1999) of continuous-time differential-algebraic equations, but many problems can also be directly modelled as implicit difference equations (Luenberger & Arbel, 1977; Pearson et al., 1988). There is a large body of work concerning the linear-quadratic optimal control problem for differential-algebraic equations, see, e.g. the studies by Mehrmann (1991), Lancaster & Rodman (1995), Backes (2006), Kurina & März (2004) and Reis et al. (2015), just to mention a few. So far, the discrete-time optimal control problem has only been discussed in a few works, most of which treat the case E = In. However, several technical assumptions are usually made. In the study by Laub (1979), the case of an invertible A with Q$$\succeq $$ 0 and R ≻ 0 is discussed. The invertibility is needed to form a symplectic matrix that is associated with a discrete-time algebraic Riccati equation \begin{align} A^{\ast}XA-X - (A^{\ast}XB + S)(B^{\ast}XB+R)^{-1} (B^{\ast}XA + S^{\ast}) + Q = 0, \quad X=X^{\ast} \end{align} (2) and the necessary optimality conditions. The invertibility condition is relaxed in the studies by Pappas et al. (1980) and Laub (1991) where instead of the symplectic matrix a symplectic matrix pencil is considered. Another difficulty arises if R is not invertible. Then, also the symplectic pencil cannot be formed and one has to turn to an extended symplectic pencil (Ionescu & Weiss, 1992), which is essentially what we will later call the BVD matrix pencil (here, BVD is an acronym for Boundary Value problem for the optimal control of Discrete systems). However, for the analysis it is still assumed that this pencil is regular. The relation of the optimal control problem to a certain linear matrix inequality (the Kalman–Yakubovich–Popov (KYP) inequality) and so-called Popov functions is discussed in the article by Stoorvogel & Saberi (1998). Here, a generalized algebraic Riccati equation is considered, where B*XB + R is not assumed to be invertible. It is shown that the solutions of this equation fulfil a certain rank-minimization property of the associated linear matrix inequality which, in contrast to the continuous-time case, do not need to be solutions of the algebraic Riccati equation (2). The case of optimal control problems for implicit difference equations, i.e. the case where E might be singular, has only been briefly considered in the literature. In the article by Bender & Laub (1987), a discrete-time algebraic Riccati equation similar to (2) is derived by transforming the system into SVD coordinates and modifying the cost functional accordingly. This analysis needs an index-one condition on the system to ensure the solvability of the optimality system. The monograph Mehrmann (1991) treats the problem numerically, i.e. structure-preserving algorithms for symplectic matrix pencils are devised. The goal of this work is a full theoretical analysis of the infinite time horizon linear-quadratic optimal control problem for implicit difference equations. In contrast to most other works, we do not impose any definiteness conditions on the cost functional nor the index of the system. Also, our notion of rank-minimality turns out to be more general than in the article by Stoorvogel & Saberi (1998). The results obtained in this paper are motivated by recent achievements for the continuous-time case (Reis, 2011; Reis et al., 2015; Voigt, 2015). This paper is structured as follows: in Section 2 we recap basic matrix and control theoretic notations and results. In Section 3 we introduce a variant of the KYP inequality for implicit difference equations given by $$ \mathcal M (P):= \left[\begin{array}{@{}cc@{}} A^{\ast}PA - E^{\ast}PE + Q &A^{\ast}PB + S \\ B^{\ast}PA + S^{\ast} &B^{\ast}PB + R \end{array}\right] \succeq{\mathcal{V}}_{(E,A,B)} 0,\qquad P=P^{\ast}, $$ a discrete-time version of the inequality introduced in the article by Reis et al. (2015), where $$\succeq _{\mathcal{V}_{(E,A,B)}}$$ denotes an inequality projected on a certain subspace $$\mathcal{V}_{(E,A,B)} $$, i.e. $$V^{\ast }\mathcal M (P)V \succeq 0$$ holds for any basis matrix V of $$\mathcal{V}_{(E,A,B)} $$. We show statements which relate the solvability of this inequality to the non-negativity of the Popov function on the unit circle, a certain rational matrix function defined by $$ \Phi(z):= \left[\begin{array}{@{}c@{}} \left(zE-A\right)^{-1}B \\ I_{m} \end{array}\right]^{\sim} \left[\begin{array}{@{}cc@{}} Q & S \\ S^{\ast} & R \end{array}\right] \left[\begin{array}{@{}c@{}} (zE-A)^{-1}B \\ I_{m} \end{array}\right] \in\mathbb{K}(z)^{m\times m}, $$ where $$G^{\sim }(z):=G\left (\overline{z}^{-1}\right )^{\ast }$$ for a rational matrix $$G(z)\in \mathbb{K}(z)^{n\times n}$$. In Sections 4 and 5 we introduce the notion of inertia for palindromic matrix pencils evaluated on the unit circle and provide spectral characterizations regarding positivity of the Popov function, similar to the characterizations which were obtained in the studies by Reis (2011) and Voigt (2015) for even matrix pencils in the continuous-time case. In Section 6 we investigate the Lur’e equation for the discrete-time optimal control problem which is a generalization of the algebraic Riccati equation (2). This means that we seek solution triples $$(X,\,K,\,L)\in \mathbb{K}^{n\times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ fulfilling $$ \left[\begin{array}{@{}cc@{}} A^{\ast}XA-E^{\ast}XE + Q & A^{\ast}XB + S \\ B^{\ast}XA + S^{\ast} & B^{\ast}XB + R \end{array}\right] ={\!}_{\mathcal{V}_{(E,A,B)}} \left[\begin{array}{@{}c@{}} K^{\ast}\\ L^{\ast} \end{array}\right] \left[\begin{array}{@{}cc@{}} K & L \end{array}\right], \quad X=X^{\ast}, $$ where $$q:=\mathrm{rk}_{\mathbb{K}_(z)} \Phi(z)$$. We show that solvability of this equation can be related to the existence of certain deflating subspaces of a palindromic matrix pencil of the form $$ z \left[\begin{array}{@{}ccc@{}} 0 & E & 0 \\ A^{\ast} & Q & S \\ B^{\ast} & S^{\ast} & R \end{array}\right] - \left[\begin{array}{@{}ccc@{}} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\right]\in\mathbb{K}[z]^{(2n+m)\times (2n+m)}. $$ The explicit construction of these deflating subspaces turns out to be much more demanding compared to the continuous-time setting. Furthermore, solutions of the Lur’e equation can be obtained from these subspaces. Due to the symmetry of the above pencil, robust and efficient structure-exploiting numerical methods (Byers et al., 2009; Schröder, 2008) can be used. Finally, in Section 7 we apply these results to the optimal control problem. Here, we consider the stabilizing solutions of the Lur’e equations which can be used to construct optimal controls (in case of existence) and to determine the optimal value. In particular, we show that under some weak conditions, the existence of stabilizing solutions is equivalent to the feasibility of the optimal control problem. We further characterize existence and uniqueness of optimal controls in terms of the zero dynamics of the closed-loop system. Finally, we discuss how the deflating subspaces of the palindromic and BVD pencils appear in the solution of the corresponding boundary value problems. Note that parts of these results are obtained in the master's thesis by Bankmann (2016). Nomenclature $$\mathbb{N}$$ = {1, 2, …}; set of natural numbers $$\mathbb{N}_{0}$$$$=\mathbb{N}\cup \{0\}$$ $$\mathbb{R}$$ field of real numbers $$\mathbb{R}^{+}$$ set of positive real numbers $$\mathbb{R}^{+}_{0}$$ set of non-negative real numbers $$\mathbb{C}$$ field of complex numbers $$\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$$ $$\mathbb{K}[z]$$ ring of polynomials with coefficients in $$\mathbb{K}$$ $$\mathbb{K}(z)$$ field of rational functions that can be expressed as fraction of elements of $$\mathbb{K}[z]$$ $${e_{i}^{k}}$$i-th unit vector in $$\mathbb{K}^{k}$$ $$K^{\mathbb{N}_{0}}$$ set of all sequences $$x=\left (x_{j}\right )_{j}$$ whose components lie in the space K $$\mathcal{R}^{m \times n}$$ set of m by n matrices with entries in a ring $$\mathcal{R}$$ $$\det A$$ determinant of a matrix $$A\in \mathbb{K}^{n \times n}$$ A* conjugate transpose of a matrix $$A\in \mathbb{K}^{m\times n}$$ A+ Moore–Penrose pseudo inverse of a matrix $$A\in \mathbb{K}^{m\times n}$$ A−* conjugate transpose of the inverse of an invertible matrix $$A\in \mathbb{K}^{n \times n}$$ $$\textrm{rk}_{\mathbb{K}(z)} A(z)$$ rank of a rational matrix $$A(z)\in \mathbb{K}[z]^{m\times n}$$ G∼ (z) $$:=G\left (\overline{z}^{-1}\right )^{\ast }$$ for a rational matrix $$G(z)\in \mathbb{K}[z]^{n\times n}$$ ∥x∥2 two-norm of a vector $$x\in \mathbb{K}^{n}$$ $$\ell ^{2}\left (\mathbb{K}^{n}\right )$$ space of quadratic-summable sequences $$x\in (\mathbb{K}^{n})^{\mathbb{N}_{0}}$$, i.e. $$\sum _{k=0}^{\infty }{\left \| x_{j}\right \|_{2}}<\infty $$ $$\| x\|_{\ell ^{2}}$$$$=\left (\sum _{k=0}^{\infty }{\left \| x_{j}\right \|^{2}_{2}}\right )^{\frac 12}$$; ℓ2-norm of a sequence $$x\in \ell ^{2}(\mathbb{K}^{n})$$ $$\Sigma_{m,n}(\mathbb{K})$$ set of all $$(E,A,B)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}$$ with regular zE − A $$\Sigma_{m,n}^w(\mathbb{K})$$ set of all $$(E, A, B, Q, S, R)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{m\times m}$$, where $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ and Q and R are Hermitian $$\mathfrak{B}_{(E,A,B)}$$ set of all $$({x,u})$$ which solve the discrete-time implicit difference equation (1) $$\mathcal{V}_{(E,A,B)} $$ system space of $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$, see Def. 2.9 $$\mathcal{W}_{(E,A,B)} $$ set of all $$x^{0}\in \mathbb{K}^{n}$$ such that there exists $$(x,u)\in \mathfrak{B}_{(E,A,B)}$$ with Ex0 = Ex0 $${\mathfrak{ZD}}_{(E,A,B,C,D)}$$ set of all $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that $$Cx_{j} + Du_{j} =0,\,j\in \mathbb{N}_{0}$$ $$\mathcal W^{0}_{(E,A,B,C,D)}$$ set of all $$x^{0}\in \mathbb{K}^{n}$$ such that there exists $$({x,u})\in {\mathfrak{ZD}}_{(E,A,B,C,D)}$$ with Ex0 = Ex0 2. Mathematical preliminaries 2.1. Matrix pencils In this section we briefly discuss some basic notions of matrix pencils $$zE-A\in \mathbb{K}[z]^{m\times n}$$. Definition 2.1 (Equivalence of matrix pencils) Two matrix pencils $$zE_{1} - A_{1}, \, zE_{2} - A_{2} \in \mathbb{K}[z]^{m\times n}$$ are called equivalent if there exist invertible matrices $$W \in \mathbb{K}^{m\times m}$$ and $$T \in \mathbb{K}^{n \times n}$$ such that $$ zE_{2} - A_{2} = W \left(zE_{1} - A_{1}\right) T. $$ Each matrix pencil $$zE-A \in \mathbb{C}[z]^{m\times n}$$ can be transformed to Kronecker canonical form via equivalence transformations. This is made precise in the next theorem. Theorem 2.2 (Kronecker canonical form (KCF)) (Gantmacher, 1960) For every matrix pencil $$zE-A\in \mathbb{C}[z]^{m\times n}$$, there exist invertible matrices $$W\in \mathbb{C}^{m\times m}$$ and $$T\in \mathbb{C}^{n\times n}$$ such that $$ W\left(zE-A\right)T = \operatorname{diag} \left( {K}_{1}(z),\,\ldots,\,{K}_{l}(z)\right),\qquad l\in\mathbb{N}, $$ where each block Kj(z) is in one of the forms in Table 1 and $$ N_{k}= \left[\begin{array}{@{}cccc@{}} 0& 1 & & \\ & \ddots&\ddots &\\ & & \ddots&1\\ &&&0 \end{array}\right]\in \mathbb{C}^{k\times k},\, {K^{R}_{k}} = \left[\begin{array}{@{}ccc@{}} 1& &\\ 0&\ddots&\\ &\ddots &1\\ &&0 \end{array}\right]\in\mathbb{C}^{(k+1)\times k},\, {K^{L}_{k}} = \left[\begin{array}{@{}cccc@{}} 0& 1 & & \\ & \ddots&\ddots &\\ & &0&1 \end{array}\right]\in\mathbb{C}^{k\times (k+1)}. $$ The KCF is unique up to permutations of the blocks. Table 1. Blocks in Kronecker canonical form Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ View Large Table 1. Blocks in Kronecker canonical form Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ View Large Equivalent matrix pencils share the same spectral structure which can be read off the KCF. Here, blocks of type K1 and K2 correspond to finite eigenvalues and infinite eigenvalues, respectively. Blocks of these types and combinations of them are regular. Blocks of types K3 and K4 are rectangular and thus not regular. Note that we allow for blocks of type K3 or K4 to have zero rows or zero columns, respectively. Such blocks represent a zero row or zero column, in the KCF of zE − A, respectively. Based on the KCF we define the index of the pencil $$zE-A \in \mathbb{C}[z]^{m\times n}$$ as the size k of the largest block of type K2 or K4 in its KCF (Berger & Reis, 2013). When characterizing the eigenstructure of matrices $$A\in \mathbb{K}^{n \times n}$$, often invariant subspaces are involved, i.e. subspaces $$\mathcal V \subseteq \mathbb{K}^{n}$$ such that $$A\mathcal V \subseteq \mathcal V$$. The generalization of invariant subspaces to matrix pencils $$zE-A\in \mathbb{K}[z]^{m\times n}$$ are so-called deflating subspaces. Here, we are using a general definition which is also suitable for singular matrix pencils, see the studies by Van Dooren (1983) and Voigt (2015). Definition 2.3 (Basis matrix, deflating subspaces) Let $$zE-A\in \mathbb{K}[z]^{n\times n}$$ and some subspace $$\mathcal{Y}\subseteq \mathbb{K}^{n}$$ be given. (a) A matrix $$Y\in \mathbb{C}^{n\times k}$$ with full column rank such that $$\mathcal{Y} = \operatorname{im} Y$$ is called basis matrix of $$\mathcal{Y}$$. (b) If for a basis matrix $$Y\in \mathbb{C}^{n\times k}$$ of $$\mathcal{Y}$$ there exist $$W\in \mathbb{C}^{n\times p}$$ and $$z\hat{E}- \hat{A}\in \mathbb{C}[z]^{p\times n}$$ such that $$ \left(zE-A\right)V = W \left(z\hat{E} - \hat{A}\right) $$ and $$\textrm{rk}_{\mathbb{C}(z)} \left (z\hat{E}- \hat{A}\right ) = p$$, then $$\mathcal{Y}$$ is called deflating subspace of zE − A. Indeed, every invariant subspace $$\mathcal V \subseteq \mathbb{K}^{n}$$ of $$A\in \mathbb{K}^{n \times n}$$ with basis matrix $$V\in \mathbb{K}^{n\times k}$$ describes a deflating subspace for the associated matrix pencil zIn − A by setting W = V and $$z\hat{E} - \hat{A}= zI_{k} -\Lambda $$, where $$\Lambda \in \mathbb{K}^{k\times k}$$ fulfils AV = VΛ. An important property that deflating subspaces might have is E-neutrality. Definition 2.4 (E-neutrality) (Gohberg et al., 2006; Reis, 2011) Let $$E\in \mathbb{K}^{n \times n}$$ and some subspace $$\mathcal{Y}\subseteq \mathbb{K}^{n}$$ be given. Then, $$\mathcal{Y}$$ is called E-neutral if for all $$x,\,y\in \mathcal{Y}$$ it holds that x*Ey = 0. It is called maximally E-neutral if every proper superspace $$\mathcal{W}\supsetneq \mathcal{Y}$$ is not E-neutral. For a subspace $$\mathcal{Y}\subseteq \mathbb{K}^{n}$$ we can check E-neutrality by testing whether Y*EY = 0, where $$Y\in \mathbb{K}^{n\times k}$$ is given such that $$ \operatorname{im} Y = \mathcal{Y}$$. 2.2. Feedback equivalence Let $$\Sigma_{m,n}(\mathbb{K}) $$ denote the set containing all the system triples $$(E,A,B)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}$$ with regular zE − A, i.e. $$\det (zE-A)\neq 0$$. Later, we will also use the set $$\Sigma_{m,n}^w(\mathbb{K}) $$ containing all systems $$(E, A, B, Q, S, R)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{m\times m}$$, where $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$ and Q and R are Hermitian. Furthermore, we call the space of all $$(\,{x,\, u}\,)\in (\mathbb{K}^{n})^{\mathbb{N}_{0}}\times (\mathbb{K}^{m})^{\mathbb{N}_{0}}$$ that solve the IDE (1) the behaviour of the system $$(E,A,B)$$. The behaviour is denoted by $$\mathfrak{B}_{(E,A,B)}$$. In this subsection we introduce an equivalence relation on the set $$\Sigma_{m,n}(\mathbb{K}) $$ which will be particularly useful in Sections 3 and 6. This subsection is mainly based on the article by Reis et al. (2015, Section 2.3). Definition 2.5 (Feedback equivalence) Two systems $$(E_{i},\,A_{i},\,B_{i})\in \Sigma_{m,n}(\mathbb{K}) ,\,i=1,2$$, are said to be feedback equivalent if there exist invertible matrices $$W,\,T\in \mathbb{K}^{n \times n}$$ and a feedback matrix $$F\in \mathbb{K}^{m\times n}$$ such that \begin{equation*} \big[zE_{2}-A_{2}\quad -B_{2}\big] = W \big[zE_{1}-A_{1}\quad -B_{1}\big]\,\mathcal{T}_{F}, \end{equation*} where $$ \mathcal{T}_{F} = \left[\begin{array}{@{}cc@{}} T & 0 \\ FT & I_{m} \end{array}\right]. $$ If this is the case we say that (E1, A1, B1) is feedback equivalent to (E2, A2, B2) via W and $$\mathcal{T}_{F}$$. Note that in the behaviour sense, i.e. looking at the system defined by $$z\mathcal{E}-\mathcal{A}$$, where $$ \mathcal{E} := [E\quad 0],\qquad \mathcal{A}:= [A\quad B], $$ feedback equivalence corresponds to strong equivalence as introduced in the book by Kunkel & Mehrmann (2006). In particular, this means that feedback equivalence is indeed an equivalence relation, (Kunkel & Mehrmann, 2006, Lemma 2.2.) Given such an equivalence relation, one is usually interested in some condensed form. The following result provides such a form. Theorem 2.6 (Feedback equivalence form) (Ilchmann & Reis, 2017, Proposition 2.12) Let the system $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$ be given. Then, $$(E,A,B)$$ is feedback equivalent to $$(E_{F},A_{F},B_{F})$$ via some W and $$\mathcal T_{F}$$, where \begin{align} [zE_{F} - A_{F}\quad -B_{F}]=\left[ \begin{array}{@{} ccc|c@{}} zI_{n_{1}}-A_{11} & 0 & 0 & -B_{1} \\ 0 & - I_{n_{2}} & zE_{23} & -B_{2} \\ 0 & 0 & zE_{33} - I_{n_{3}} & 0 \end{array}\right], \end{align} (3)$$n_{1}, n_{2}, n_{3}\in \mathbb{N}_{0}$$ and E33 is nilpotent. A similar form has also been achieved in the article by Byers et al. (1997, Theorem 4.1) via unitary transformations. Example 2.7 Consider the system given by \begin{align} E= \left[\begin{array}{@{}cc@{}} 0 & 0\\ 0 & 1 \end{array}\right],\quad A= \left[\begin{array}{@{}cc@{}} -1 & 1 \\ 1 & 0\\ \end{array}\right],\quad B= \left[\begin{array}{@{}c@{}} -1\\ 0 \end{array}\right]. \end{align} (4) We obtain that the system is feedback equivalent to \begin{align} [z E_{F} - A_{F}\quad -B_{F}]= W \left[z E- A\quad -B\right]\mathcal T_{F}=\left[\begin{array}{@{}ccc@{}} z -1 & 0 & 1\\ 0 & -1 & -1 \end{array}\right] \end{align} (5) via zero feedback, i.e. F = 0 and $$ W= \left[\begin{array}{@{}cc@{}} 1 & 1\\ -1 & 0 \end{array}\right], \quad\mathcal T_{F}= \left[\begin{array}{@{}ccc@{}} 1 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{array}\right]. $$ Thus, we have n1 = n2 = 1 and n3 = 0 in (3). Proposition 2.8 Let the system $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$ be feedback equivalent to the system $$(E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ in feedback equivalence form (3). Further, denote by $$ (I_{n_{1}},A_{11},B_{1})\in \Sigma_{m,n1}(\mathbb{K}) $$ the associated explicit difference equation (EDE) system. Then, for $$\lambda \in \mathbb{C}$$ we have $$\det \left (\lambda E_{F} -A_{F}\right )\neq 0$$ if and only if $$\det \left (\lambda I_{n_{1}} - A_{11}\right )\neq 0$$. Proof. Note that every nilpotent matrix $$E_{33}\in \mathbb{K}^{n_{3}\times n_{3}}$$ has only eigenvalues zero and thus $$\det \left (zE_{33}-I_{n_{3}}\right ) = (-1)^{n_{3}}$$. Then, the assertion follows immediately from the block-triangular structure of zEF − AF. 2.3. System space In this subsection we investigate properties of the solution space of the IDEs given by a system $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$. This section is based on the article by Reis et al. (2015, Chapter 3). Definition 2.9 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$. The smallest subspace $$\mathcal{V}_{(E,A,B)} \subseteq \mathbb{K}^{n+m}$$ such that $$ \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right) \in \mathcal{V}_{(E,A,B)} $$ for all $$j\in \mathbb{N}_{0}$$ and for all $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ are called the system space of $$(E,A,B)$$. Lemma 2.10 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$. Further, assume that $$(E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is feedback equivalent to $$(E,A,B)$$ via W and $$\mathcal{T}_{F}$$. Then, the system spaces $$\mathcal{V}_{(E,A,B)} $$ and $$\mathcal{V}_{(E_F,A_F,B_F)}$$ of $$(E, A, B)$$ and $$(E_{F},A_{F},B_{F})$$, respectively, are related via $$ \mathcal{V}_{(E,A,B)} = \mathcal{T}_{F} \mathcal{V}_{(E_F,A_F,B_F)}. $$ Proof. The assertion has been shown in the article by Reis et al. (2015, Lemma 3.2). Proposition 2.11 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ be given. Further, assume that $$(E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is feedback equivalent to $$(E, A, B)$$ via W and $$\mathcal{T}_{F}$$ such that $$(E_{F},A_{F},B_{F})$$ is in feedback equivalence form (3). Then, we have: (a) It holds that $$\mathcal{V}_{(E_F,A_F,B_F)} = \operatorname{im} V_{F}$$, where \begin{align} V_{F} := \left[\begin{array}{@{}cccc@{}} I_{n_{1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & -B_{2}\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{m} \end{array}\right]\in\mathbb{K}^{(n+m)\times (n+m)}. \end{align} (6) (b) It holds that \begin{align} [A \ \ B] \left[\begin{array}{@{}c@{}} \left(zE-A\right)^{-1}B\\ I_{m} \end{array}\right] = z \ \left[E \ \ 0\right]\left[\begin{array}{@{}c@{}} (zE-A)^{-1}B\\ I_{m} \end{array}\right]. \end{align} (7) (c) For all $$\lambda \in \mathbb{C}$$ with $$\det (\lambda E - A)\neq 0 $$ it holds that $$ \operatorname{im} \left[\begin{array}{@{}c@{}} (\lambda E-A)^{-1}B\\ I_{m} \end{array}\right] \subseteq \mathcal{V}_{(E,A,B)} . $$ (d) Consider VF as in (a) and let $$V:= \mathcal T_{F} V_{F}$$. Then, $$ V_{F} \mathcal{V}_{(E_F,A_F,B_F)}= \mathcal{V}_{(E_F,A_F,B_F)} $$ and $$ V \mathcal T_{F}^{-1} \mathcal{V}_{(E,A,B)} = \mathcal{V}_{(E,A,B)}. $$ Proof. Assertion (a) is shown in the proof of Reis et al. (2015, Proposition 3.3). Assertion (c) is shown in the study by Reis et al. (2015, Lemma 3.5), where part (b) is obtained in the proof of Reis et al. (2015, Lemma 3.5). For part (d) see the study by Bankmann (2016, Proposition 2.29(d)). 2.4. Controllability and asymptotic stability Before we introduce the linear-quadratic optimal control problem, we first need to recap several concepts of controllability and asymptotic stability for the system given by $$(E, A, B)\in \Sigma_{m,n}(\mathbb{K})$$. These concepts are similar to the continuous-time case as in the studies by Dai (1989) and Bunse-Gerstner et al. (1999) and are discussed in, e.g. the studies by Dai (1989) and Stykel (2003). Definition 2.12 The system $$(E, A, B)\in \Sigma_{m,n}(\mathbb{K}) $$ or $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ is called (a) completely controllable (C-controllable) if for every initial point $$x^{0}\in \mathbb{K}^{n}$$ and every final point $$x^{\textrm{f}}\in \mathbb{K}^{n}$$ there exists a $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that x0 = x0 and $$x_{j_{\textrm{f}}}=x^{\textrm{f}}$$ at some timepoint $$j_{\textrm{f}}\in \mathbb{N}_{0}$$; (b) controllable on the reachable set (R-controllable) if for every initial point $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and every final point $$x_{\textrm{f}}\in \mathcal{W}_{(E,A,B)} $$ there exist $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0 and $$Ex_{j_{\textrm{f}}}=Ex^{\textrm{f}}$$ at some timepoint $$j_{\textrm{f}}\in \mathbb{N}_{0}$$; (c) I-controllable if for every initial point $$x^{0}\in \mathbb{K}^{n}$$ there exists $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0, i.e. $$\mathcal{W}_{(E,A,B)} =\mathbb{K}^{n}$$; (d) stabilizable if for every initial point $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ there exists a $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0 and $$\lim \nolimits_{j\to \infty } Ex_{j} = 0$$. In the case where E = In, the notions R-controllability and C-controllability coincide and thus for systems of the form $$(I_{n}, A, B)\in \Sigma_{m,n}(\mathbb{K})$$ we omit the prefix R or C and say that they are controllable. Table 2 shows well-known characterizations of the different controllability notions (Dai, 1989; Stykel, 2003; Berger, 2014). Table 2. Algebraic characterizations of controllability and stabilizability, where $$S_{\infty }$$ is a basis matrix of $$\ker E$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ View Large Table 2. Algebraic characterizations of controllability and stabilizability, where $$S_{\infty }$$ is a basis matrix of $$\ker E$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ View Large Eigenvalues $$\lambda \in \mathbb{C}$$ of $$zE-A\in \mathbb{K}[z]^{n\times n}$$ such that $$ \operatorname{rk} \left [\lambda E -A \quad B \right ] \neq n$$ destroy the controllability property. In this case we thus say that there is an uncontrollable mode at λ; otherwise we say that there is a controllable mode at λ. Lemma 2.13 Let the system $$(E, A, B)\in \Sigma_{m,n}(\mathbb{K})$$ be feedback equivalent to the system $$(E_{F}, A_{F}, B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ in feedback equivalence form (3) via W and $$\mathcal T_{F}$$. Furthermore, denote by $$(I_{n_{1}},A_{11},B_{1})\in \Sigma_{m,n1}(\mathbb{K})$$ the associated EDE system. Then, we have: (a) Let $$\lambda \in \mathbb{C}$$. Then, the system $$(E_{F},A_{F},B_{F})$$ has an uncontrollable mode at λ if and only if the system $$(I_{n_{1}}, A_{11}, B_{1})$$ has an uncontrollable mode at λ. (b) The system $$(E_{F},A_{F},B_{F})$$ is R-controllable if and only if $$(I_{n_{1}}, A_{11}, B_{1})$$ is controllable. (c) The system $$(E, A, B)$$ is I-controllable if and only if W and $$\mathcal T_{F}$$ can be chosen such that for $$(E_{F},A_{F},B_{F})$$ it holds that n3 = 0. Proof. Assertion (a) is shown in the study by Reis et al. (2015, Lemma 2.9(c)). Then, assertion (b) is an immediate consequence of (a), since by Table 2 the system $$(E_{F},A_{F},B_{F})$$ is R-controllable if and only if all $$\lambda \in \mathbb{C}$$ there are controllable modes of $$(E_{F},A_{F},B_{F})$$ at λ. Part (c) is shown in the study by Ilchmann & Reis (2017, Proposition 2.12). 2.5. Zero dynamics In this subsection we consider IDEs with an output of the form \begin{align} E\sigma x_{j} = Ax_{j} + Bu_{j}, \quad y_{j} = Cx_{j} + Du_{j}, \end{align} (8) where $$C \in \mathbb{K}^{q \times n}$$ and $$D \in \mathbb{K}^{q \times m}$$. The set of such systems with $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ is denoted by $$\Sigma _{m,n,q}(\mathbb{K})$$ and we write $$(E,A,B,C,D) \in \Sigma _{m,n,q}(\mathbb{K})$$. The zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ of (8) simply consists of all $$(x,u) \in \mathfrak{B}_{(E,A,B)}$$ that result in a zero output, i.e. $$ {\mathfrak{ZD}}_{(E,A,B,C,D)} := \left\{ (x,u) \in \left(\mathbb{K}^{n}\right)^{\mathbb{N}_{0}} \times \left(\mathbb{K}^{m}\right)^{\mathbb{N}_{0}} \; \bigg| \; \left[\!\!\begin{array}{cc} E \sigma - A & -B \\ C & D \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{j} \\ u_{j} \end{array}\!\!\right) = 0,\, j = 0,\,1,\,2,\,\ldots \right\}\!. $$ The set of zero dynamics with ‘initial state’ $$x^{0} \in \mathbb{K}^{n}$$ is defined by $$ {\mathfrak{ZD}}_{(E,A,B,C,D)}\big(x^{0}\big) := \left\{ (x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} \; \Big| \; Ex_{0} = Ex^{0} \right\}. $$ The set of consistent initial shift variables for the zero dynamics is given by $$ \mathcal{W}^{0}_{(E,A,B,C,D)} := \left\{ x^{0} \in \mathbb{K}^{n} \; \Big| \; {\mathfrak{ZD}}_{(E,A,B,C,D)}(x^{0}) \neq \emptyset \right\}. $$ The following definition is an adaptation of the definition for continuous-time systems, see the studies by Voigt (2015) and Ilchmann & Reis (2017). Definition 2.14 The zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ with set of consistent initial shift variables $$\mathcal{W}^{0}_{(E,A,B,C,D)} $$ is called (a) stabilizable, if for all $$x^{0} \in \mathcal{W}^{0}_{(E,A,B,C,D)} $$, there exists an $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} \left (x^{0}\right )$$ such that $$\lim _{j \to \infty } \left (x_{j},u_{j}\right ) = 0$$; (b) asymptotically stable, if for all $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} $$ it holds $$\lim _{j \to \infty } \left (x_{j},u_{j}\right ) = 0$$; (c) strongly stabilizable, if it is stabilizable and $$\mathcal{W}^{0}_{(E,A,B,C,D)} = \mathbb{K}^{n}$$; (d) strongly asymptotically stable, if it is asymptotically stable and $$\mathcal{W}^{0}_{(E,A,B,C,D)} = \mathbb{K}^{n}$$. Proposition 2.15 Let $$(E,A,B,C,D) \in \Sigma _{m,n,q}(\mathbb{K})$$ be given and define $$\mathcal{R}(z) := \left[{{zE - A} \atop {C}}\quad {{-B} \atop {D}}\right] \in \mathbb{K}[z]^{(n+q) \times (n+m)}$$. Then, the zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ with the space of consistent shift variables $$\mathcal{W}^{0}_{(E,A,B,C,D)} $$ is (a) stabilizable, if and only if $$ \operatorname{rk}_{\mathbb{K}(z)} \mathcal{R}(z) = \operatorname{rk} \mathcal{R}(\lambda )$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$; (b) stabilizable, if and only if for all $$x^{0} \in \mathcal{W}^{0}_{(E,A,B,C,D)} $$ there exists a $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} \left (x^{0}\right )$$ such that $$\lim _{j \to \infty } Ex_{j} = 0$$; (c) asymptotically stable, if and only if $$ \operatorname{rk} \mathcal{R}(\lambda ) = n+m$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq 1 $$; (d) asymptotically stable, if and only if for all $$x^{0} \in \mathcal{W}^{0}_{(E,A,B,C,D)} $$ there exists a unique $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} (x^{0})$$ such that $$\lim _{j \to \infty } Ex_{j} = 0$$; (e) strongly stabilizable, if and only if $$ \operatorname{rk}_{\mathbb{K}(z)} \mathcal{R}(z) = \operatorname{rk} \mathcal{R}(\lambda )$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$ and the index of $$\mathcal{R}(z)$$ is at most one; (f) strongly asymptotically stable, if and only if $$ \operatorname{rk} \mathcal{R}(\lambda ) = n+m$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$ and the index of $$\mathcal{R}(z)$$ is at most one; Proof. The proof follows the lines of the proof of Ilchmann & Reis (2017, Prop. 4.3). It is analogously verified that the zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ is (a) stabilizable, if and only if all blocks of type K1 in the KCF of $$\mathcal{R}(z)$$ correspond to eigenvalues with |λ| < 1; (b) asymptotically stable, if and only if all blocks of type K1 in the KCF of $$\mathcal{R}(z)$$ correspond to eigenvalues with |λ| < 1 and the number of blocks of type K3 in the KCF of $$\mathcal{R}(z)$$ is zero. Together with the results in Ilchmann & Reis (2017, Rem. 2.5 (d), (e)), this shows statements (a)–(d). Moreover, it is checked that $$\mathcal{W}^{0}_{(E,A,B,C,D)} = \mathbb{K}^{n}$$ is equivalent to the condition that for every $$x^{0} \in \mathbb{K}^{n}$$ the IDE $$ \left[\!\!\begin{array}{cc} E & 0 \\ 0 & 0 \end{array}\!\!\right]\sigma \left(\!\!\begin{array}{c} x_{j} \\ u_{j} \end{array}\!\!\right) = \left[\!\!\begin{array}{cc} A & B \\ C & D \end{array}\!\!\right]\left(\!\!\begin{array}{c} x_{j} \\ u_{j} \end{array}\!\!\right), \quad Ex_{0} = Ex^{0} $$ has a solution. This is equivalent to $$\mathcal{R}(z)$$ being of index at most one (Berger, 2014) and shows statements (e) and (f). 2.6. Linear-quadratic optimal control One main goal of this work is to provide tools for analysing the discrete-time infinite horizon linear-quadratic control problem (Mehrmann, 1991; Kurina & März, 2004; Backes, 2006). It is given by the following: For $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ find $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0, $$\lim \limits _{j\rightarrow \infty } Ex_{j}= 0$$ and the objective function \begin{align} \mathcal{J}(x,u) := \sum_{j=0}^{\infty}{ \left(\!\!\begin{array}{c} x_{j}\\ u_{j} \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} Q & S\\ S^{\ast} & R \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{j}\\ u_{j} \end{array}\!\!\right) } \end{align} (9) is minimized. In other words, we are interested in the value of the functional $$\mathcal{W}_{+}(Ex^0) :E\mathcal{W}_{(E,A,B)} \rightarrow \mathbb{R}^{+}_{0}\cup \left \{\pm \infty \right \}$$ defined by $$ \mathcal{W}_{+}(Ex^0) := \inf \left\{ \mathcal{J}(x,u) \,\middle|\, (x,u)\in \mathfrak{B}_{(E,A,B)},\, Ex_{0}=Ex^{0},\,\lim\limits_{j\rightarrow\infty} Ex_{j}= 0 \right\}. $$ The problem is called feasible if $$\infty>\mathcal{W}_{+}(Ex^0) >-\infty $$. It is called solvable if the infimum is actually a minimum. Note that for $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ the existence of $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0 is guaranteed by the definition of $$\mathcal{W}_{(E,A,B)} $$. If further $$(E, A, B)$$ is stabilizable we can choose u such that in addition $$\lim \nolimits _{j\rightarrow \infty } Ex_{j}= 0$$, i.e. $$\mathcal{W}_{+}(Ex^0) < \infty $$. It can be easily seen that the objective function $$\mathcal{J}(x,u) $$ does not change if the system $$(E_{F}, A_{F}, B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is equivalent to $$(E, A, B)$$ via W and $$\mathcal{T}_{F}$$ and we use the modified weights $$ \left[\!\!\begin{array}{cc} Q_{F} & S_{F}\\ S_{F}^{\ast}& R_{F} \end{array}\!\!\right]:= \left[\!\!\begin{array}{cc} T^{\ast}\left(Q + F^{\ast}S^{\ast} + SF + F^{\ast}RF\right)T & T^{\ast}\left(S+F^{\ast}R\right)\\ \left(S^{\ast}+RF\right)T & R \end{array}\!\!\right]. $$ If we assume that the system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ is I-controllable and that \begin{align} \left[\!\!\begin{array}{cc} Q & S\\ S^{\ast}& R \end{array}\!\!\right]\succeq 0 \end{align} (10) then it is well-known that in this case solutions of the optimal control problem can be characterized via certain structured matrix pencils, see the studies by Mehrmann (1991), Mackey et al. (2006), Kunkel & Mehrmann (2008) and Byers et al. (2009). One main contribution of this work is that we actually can drop these assumptions. In the discrete-time case, applying Pontryagin’s maximum principle (Pontryagin et al., 1962; Mehrmann, 1991) leads to \begin{align} \left[\!\!\begin{array}{ccc} 0 & E & 0 \\ A^{\ast} & 0 & 0 \\ B^{\ast} & 0 & 0 \end{array}\!\!\right] \sigma \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right) = \left[\!\!\begin{array}{ccc} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\!\!\right] \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right),\quad Ex_{0} = E x^{0}, \quad \lim\limits_{j\to\infty}{E^{\ast}\mu_{j}}=0, \end{align} (11) where $$(x,u)\in \mathfrak{B}_{(E,A,B)}$$, $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and $$\mu \in{(\mathbb{K}^{n})^{\mathbb{N}_{0}}}$$ denote some Lagrange multipliers. This IDE can be analysed by means of the matrix pencil \begin{align} z\mathcal{E}-\mathcal{A} = \left[\!\!\begin{array}{ccc} 0 & zE-A & -B \\ zA^{\ast}-E^{\ast} & -Q & -S \\ zB^{\ast} & -S^{\ast} & -R \end{array}\!\!\right]\in\mathbb{K}[z]^{(2n+m)\times (2n+m)}, \end{align} (12) a so-called BVD-pencil. The structure of this pencil is not invariant under unitary transformations which leads to problems in the numerical treatment (Byers et al., 2009). In the studies by Schröder (2008) and Byers et al. (2009) it is shown how we can achieve a more structured version if we introduce new variables $$ m_{j}:=\mu_{j} - \mu_{j+1}. $$ This reformulation yields \begin{align} \left[\!\!\begin{array}{ccc} 0 & E & 0 \\ A^{\ast} & Q & S \\ B^{\ast} & S^{\ast} & R \end{array}\!\!\right] \sigma \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right) = \left[\!\!\begin{array}{ccc} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\!\!\right] \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right),\quad Ex_{0} = E x^{0}, \quad \sum\limits_{j=0}^{\infty}{E^{\ast}m_{j}}=E^{\ast}\mu_{j} \end{align} (13) with the corresponding matrix pencil \begin{align} z\mathcal{E}-\mathcal{A} = z\mathcal{A}^{\ast}-\mathcal{A} \left[\!\!\begin{array}{ccc} 0 & zE-A & -B \\ zA^{\ast}-E^{\ast} & (z-1)Q & (z-1)S \\ zB^{\ast} & (z-1)S^{\ast} & (z-1)R \end{array}\!\!\right]\in\mathbb{K}[z]^{(2n+m)\times (2n+m)}. \end{align} (14) This pencil has the special property of being palindromic, i.e. $$\mathcal{E}=\mathcal{A}^{\ast }$$. This structure is preserved under congruence transformation and there exist numerically stable and structure-preserving methods for the computation of eigenvalues and deflating subspaces. In particular, simple eigenvalues on the unit circle stay on the unit circle (Schröder, 2008; Byers et al., 2009). In Section 4 we discuss properties of palindromic pencils in more detail. However, in the article by Mehrmann & Scholz (2014) it is shown that in an abstract Banach space setting the operator associated to the palindromic pencil (14) is not self-adjoint; in contrast to the operator associated to the so-called even pencil arising in continuous-time, see also the article by Kunkel et al. (2014). We show in Sections 6 and 7 that in analogy to the continuous-time case study by Reis et al. (2015), also in the discrete-time case, we can drop assumption (10) to obtain the necessary optimality conditions (11) and (13). 3. Kalman-Yakubovich-Popov lemma Consider the weighted system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ and corresponding system space $$\mathcal{V}_{(E,A,B)} $$. In this section we relate positive semi-definiteness on the unit circle of the Popov function—a specific rational matrix function—to the solvability of a certain matrix inequality, namely the inequality. We will see in Section 7 that positive semi-definiteness on the unit circle of the Popov function is necessary for feasibility of the optimal control problem (9). Definition 3.1 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given. Consider $$P=P^{\ast }\in \mathbb{K}^{n \times n}$$ and \begin{align} \mathcal{M}(P):= \left[\!\!\begin{array}{cc} A^{\ast}PA - E^{\ast}PE + Q & A^{\ast}PB + S \\ B^{\ast}PA + S^{\ast} & B^{\ast}PB + R \end{array}\!\!\right]. \end{align} (15) If $$\mathcal{M}(P)\succeq{}_{\mathcal{V}_{(E,A,B)}}0$$, then P is called solution of the discrete-time KYP inequality \begin{align} \mathcal{M}(P)\succeq{}_{\mathcal{V}_{(E,A,B)}} 0,\qquad P^{\ast}=P. \end{align} (16) Throughout this section we will make use of the system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ being transformed to $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ via feedback equivalence, i.e. we have invertible $$W,\,T\in \mathbb{K}^{n \times n}$$ and a feedback matrix $$F\in \mathbb{K}^{m\times n}$$ such that \begin{align} E_{F}=&\, WET,\quad A_{F}=W(A+BF)T,\quad B_{F}=WB,\nonumber \\ Q_{F}=&\, T^{\ast}\left(Q+SF+F^{\ast}S^{\ast}+F^{\ast}RF\right)T,\quad S_{F}=T^{\ast}\left(S+F^{\ast}R\right),\quad R_{F}=R. \end{align} (17) These transformations will allow us to extract an EDE formulation from the IDE problem. The next results are crucial for the proof of the KYP lemma in the IDE case and are mainly adaptions of the corresponding results in Reis et al. (2015, Section 4). Lemma 3.2 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$. Then, we have $$ \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} A^{\ast}PA-E^{\ast}PE & A^{\ast}PB\\ B^{\ast}PA & B^{\ast}PB \end{array}\!\!\right] \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right] = 0. $$ Proof. By using (7) we obtain \begin{align*} &\left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} A^{\ast}PA-E^{\ast}PE & A^{\ast}PB\\ B^{\ast}PA & B^{\ast}PB \end{array}\!\!\right] \left[\!\! \begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right] \\ &\quad \overset{\hphantom{\eqref{eq:sysspace}}}= \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left( \left[\!\!\begin{array}{c} A^{\ast}\\ B^{\ast} \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} A & B \end{array}\!\!\right] - \left[\!\!\begin{array}{c} E^{\ast}\\ 0 \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} E & 0 \end{array}\!\!\right] \right) \left[ \begin{array}{@{}c@{}} (zE-A)^{-1}B \\ I_{m} \end{array} \right] \\ &\quad \overset{(7)}= \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left( \overline{z}^{-*} \left[\!\!\begin{array}{c} E^{\ast}\\ 0 \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} E & 0 \end{array}\!\!\right] z - \left[\!\!\begin{array}{c} E^{\ast}\\ 0 \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} E & 0 \end{array}\!\!\right] \right) \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right] = 0. \end{align*} Lemma 3.3 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ with corresponding system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form as in (17) be given. Further, let $$P=P^{\ast }\in \mathbb{K}^{n \times n}$$ and set PF = W−*PW−1 and $$ \mathcal{T}_{F}= \left[\!\!\begin{array}{cc} T & 0 \\ FT& I_{m} \end{array}\!\!\right]. $$ Then, $$ \mathcal{M}_{F}(P_{F})= \mathcal{T}_{F}^{\ast } \mathcal{M}(P) \mathcal{T}_{F}, $$ where $$\mathcal{M}_{F}(P_{F})$$ is the matrix in (15) with respect to $$(E_{F}, A_{F}, Q_{F}, S_{F}, R_{F})$$. For the generalization of the KYP inequality to IDEs we first need to understand relations between the different Popov functions and KYP inequalities corresponding to systems $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ and $$ (E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) and how they are related to EDEs. If $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ is also in feedback equivalence form (3), then the associated EDE part is given by $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K})$$ which is defined by \begin{align} A_{s}=&\, A_{11},\quad B_{s}=B_{1},\nonumber\\ Q_{s}=&\, Q_{11},\quad S_{s}=S_{1}-Q_{12}B_{2},\quad R_{s}=B_{2}^{\ast}Q_{22}B_{2} - B_{2}^{\ast}S_{2}-S_{2}^{\ast}B_{2}+R. \end{align} (18) Proposition 3.4 Consider the Popov function $$ \Phi_{F}(z) := \left[\!\!\begin{array}{c} \left(zE_{F}-A_{F}\right)^{-1}B_{F} \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} Q_{F} & S_{F} \\ S_{F}^{\ast} & R_{F} \end{array}\!\!\right] \left[\!\!\begin{array}{c} (zE_{F}-A_{F})^{-1}B_{F} \\ I_{m} \end{array}\!\!\right] \in \mathbb{K}[z]^{m\times m} $$ of the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17). (a) The Popov functions ΦF(z) and Φ(z) are related via $$ \Phi_{F}(z)=\Theta_{F}^{\sim}(z)\Phi(z)\Theta_{F}(z), $$ where $$\Theta _{F}(z)=I_{m}+FT(zE_{F}-A_{F})^{-1}B_{F}\in \mathbb{K}[z]^{m\times m}$$ is invertible. (b) Further, assume that $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ is given in feedback equivalence form as in (3) and partitioned accordingly. Then, it holds that ΦF(z) = Φs(z), where Φs(z) is the Popov function corresponding to the EDE part $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K})$$ as in (18). Proof. See the studies by Reis et al. (2015), Voigt (2015) and Bankmann (2016, Proposition 3.8). We now turn to a reduction of the problem for systems $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ in feedback equivalence form as in (17) to the corresponding EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18). Lemma 3.5 Assume that $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) is given in feedback equivalence form as in (3) and partitioned accordingly. Further, consider the corresponding EDE part $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18) and partition the Hermitian matrix $$ P_{F}= \left[\!\!\begin{array}{ccc} P_{11} & P_{12} & P_{13}\\ P_{12}^{\ast} & P_{22} & P_{23}\\ P_{13}^{\ast} & P_{23}^{\ast} & P_{33} \end{array}\!\!\right] \in\mathbb{K}^{n \times n} $$ accordingly. Then, $$P_{11}\in \mathbb{K}^{n_{1}\times n_{1}}$$ is a solution of the KYP inequality (16) corresponding to the EDE part $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)$$ if and only if PF is a solution of the KYP inequality (16) corresponding to $$ (E_F, A_F, B_F, Q_F, S_F, R_F)$$. Proof. We have \begin{align} &\left[\!\!\begin{array}{cc} A_{F}^{\ast}P_{F}A_{F} - E_{F}^{\ast}P_{F}E_{F} +Q_{F} & A_{F}^{\ast}P_{F}B_{F} +S_{F} \\ B_{F}^{\ast}P_{F}A_{F} +S_{F}^{\ast} & B_{F}^{\ast}P_{F}B_{F} +R_{F} \end{array}\!\!\right]\nonumber\\ &\quad=\left[ \begin{array}{@{}ccc|c@{}} A_{11}^{\ast}P_{11}A_{11}-P_{11} & A_{11}^{\ast}P_{12} & M_{13} & A_{11}^{\ast}P_{11}B_{1} + A_{11}^{\ast}P_{12}B_{2} \\ P_{12}^{\ast}A_{11} & P_{22} & M_{23} & P_{12}^{\ast}B_{1} + P_{22}B_{2} \\ M_{13}^{\ast} & M_{23}^{\ast} & M_{33} & M_{34} \\\hline B_{1}^{\ast}P_{11}A_{11} + B_{2}^{\ast}P_{12}^{\ast}A_{11} & B_{1}^{\ast}P_{12} +B_{2}^{\ast}P_{22} & M_{34}^{\ast} & M_{44} \end{array}\right] \nonumber\\ &\qquad\quad+ \left[ \begin{array}{@{}ccc|c@{}} Q_{11} & Q_{12} & Q_{13} & S_{1} \\ Q_{12}^{\ast} & Q_{22} & Q_{23} & S_{2} \\ Q_{13}^{\ast} & Q_{23}^{\ast} & Q_{33} & S_{3} \\ \hline S_{1}^{\ast} & S_{2}^{\ast} & S_{3}^{\ast} & R \end{array}\right] \end{align} (19) for some $$M_{13}\in \mathbb{K}^{n_{1}\times n_{3}}$$, $$M_{23}\in \mathbb{K}^{n_{2}\times n_{3}}$$, $$M_{33}\in \mathbb{K}^{n_{3}\times n_{3}}$$, $$M_{34}\in \mathbb{K}^{n_{3}\times m}$$ and $$ M_{44}=B_{1}^{\ast}P_{11}B_{1}+B_{1}^{\ast}P_{12}B_{2} +B_{2}^{\ast}P_{22}B_{2} + B_{2}^{\ast}P_{12}^{\ast}B_{1}\in\mathbb{K}^{m\times m}. $$ Let $$({x^{*},u^{*}})^{*}\in \mathcal{V}_{(E_F,A_F,B_F)} $$. Thus, by (6) there exists an $$x _{1}\in \mathbb{K}^{n_{1}}$$ such that $$ x=\left(\begin{array}{@{}c@{}} x_{1}\\ -B_{2}u\\ 0_{n_{3}\times1} \end{array}\right). $$ Then, we obtain \begin{multline*} \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{F}^{\ast}P_{F}A_{F} - E_{F}^{\ast}P_{F}E_{F} + Q_{F} & A_{F}^{\ast}P_{F}B_{F} + S_{F} \\ B_{F}^{\ast}P_{F}A_{F} + S_{F}^{\ast} & B_{F}^{\ast}P_{F}B_{F} + R_{F} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)\\ \qquad= \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{s}^{\ast}P_{11}A_{s}-P_{11}+Q_{s} & A_{s}^{\ast}P_{11}B_{s} + S_{s} \\ B_{s}^{\ast}P_{11}A_{s} + S_{s}^{\ast} & B_{s}^{\ast}P_{11}B_{s} + R_{s} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right). \end{multline*} Thus, $$ \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{s}^{\ast}P_{11}A_{s}-P_{11}+Q_{s} & A_{s}^{\ast}P_{11}B_{s} + S_{s} \\ B_{s}^{\ast}P_{11}A_{s} + S_{s}^{\ast} & B_{s}^{\ast}P_{11}B_{s} + R_{s} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right) \ge 0 $$ for all $$({x_{1},u})\in \mathbb{K}^{n_{1}+m}$$ if and only if $$ \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{F}^{\ast}P_{F}A_{F} - E_{F}^{\ast}P_{F}E_{F} + Q_{F} & A_{F}^{\ast}P_{F}B_{F} + S_{F} \\ B_{F}^{\ast}P_{F}A_{F} + S_{F}^{\ast} & B_{F}^{\ast}P_{F}B_{F} + R_{F} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)\ge0 $$ for all $$({x^{*},u^{*}})^{*}\in \mathcal{V}_{(E_F,A_F,B_F)}$$. Hence, P11 is a solution of the KYP inequality (16) corresponding to the EDE part if and only if PF solves (16) corresponding to $$(E_{F},A_{F},B_{F})$$. We are now ready to state the generalization of the KYP lemma for IDEs. Theorem 3.6 (KYP lemma for IDEs) Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given with corresponding Popov function $$\Phi (z)\in \mathbb{K}[z]^{m\times m}$$. (a) If there exists some $$P\in \mathbb{K}^{n \times n}$$ that is a solution of (16), then $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right ) \succeq 0$$ for all $$\omega \in \mathbb{R}$$ with $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. (b) If on the other hand (E, A, B) is R-controllable and $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right ) \succeq 0$$ for all $$\omega \in \mathbb{R}$$ with $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$, then there exists a solution $$P\in \mathbb{K}^{n \times n}$$ of (16). Proof. We first show assertion (a). Assume that $$P\in \mathbb{K}^{n \times n}$$ fulfils the KYP inequality (16), i.e. $$\mathcal{M}(P)\succeq_{\,\mathcal{V}_{(E,A,B)} }\!0$$. Further, let $$\omega \in \mathbb{R}$$ be such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. Then, by Lemma 3.2, together with Proposition 2.11(b), statement (a) follows due to \begin{align*} \Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right)=& \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} Q & S \\ S^{\ast} & R \end{array}\!\!\right] \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right]\\ =& \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\ast} \mathcal{M}(P) \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right] \succeq 0. \end{align*} For part (b) assume that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all $$\omega \in \mathbb{R}$$ with $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. For the system in feedback equivalence form $$(E_{F},A_{F},B_{F},Q_{F},S_{F},R_{F}) \in \Sigma^w_{m,n}(\mathbb{K})$$ and corresponding Popov function $$\Phi _{F}(z)\in \mathbb{K}[z]^{m\times m}$$ we obtain from Proposition 3.4(b) that $$\Phi _{F}\left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all $$\omega \in \mathbb{R}$$ also fulfilling $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E_{F}-A_{F}\right )\neq 0$$. In particular, by Proposition 2.8 for such ω we have $${\det\!\left(\mathrm{e}^{\mathrm{i}\omega} I_{n_{1}}-A_{11}\right )\neq 0}$$. Furthermore, by Proposition 2.13(b) the associated EDE system $$(I_{n_{1}}, A_{11}, B_{1})$$ is controllable. This means we are in the situation of this theorem for EDEs (Rantzer, 1996) for the EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18). Thus, applying Lemma 3.5 gives a solution PF of the KYP inequality (16) corresponding to the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$. Then, using Lemma 3.3 completes the proof. Example 3.7 (Example 2.7 revisited) Consider the system $$(E, A, B)$$ as in Example 2.7. From its feedback equivalence form as in (5) we obtain with (6) that $$ V_{F}= \left[\!\!\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & -1\\ 0 & 0 & 1 \end{array}\!\!\right] $$ spans the system space $$\mathcal{V}_{(E_F,A_F,B_F)}$$ and thus $$ \mathcal{V}_{(E,A,B)} = \mathcal T_{F} \mathcal{V}_{(E_F,A_F,B_F)}= \operatorname{im} \left[\!\!\begin{array}{ccc} 1 & 0 & -1\\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\!\!\right]\quad \textrm{with}\quad \mathcal T_{F}= \left[\!\!\begin{array}{ccc} 1 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{array}\!\!\right]. $$ From $$ \left[\!\!\begin{array}{cc} Q & S\\ S^{\ast} & R \end{array}\!\!\right]=I_{3} $$ we obtain as modified weights $$ \left[\!\!\begin{array}{cc} Q_{F} & S_{F}\\ S_{F}^{\ast}& R_{F} \end{array}\!\!\right]= \left[\!\!\begin{array}{cc} T^{\ast}\left(Q + F^{\ast}S^{\ast} + SF + F^{\ast}RF\right)T & T^{\ast}\left(S+F^{\ast}R\right)\\ \left(S^{\ast}+RF\right)T & R \end{array}\!\!\right] = \left[\!\!\begin{array}{ccc} 2 & 1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{array}\!\!\right]. $$ Moreover, the associated EDE part as in (18) is given by \begin{align} A_{s}=1,\quad B_{s}=-1,\quad Q_{s}= 2,\quad S_{s}=-1,\quad R_{s}=2. \end{align} (20) Thus, P11 solves the KYP inequality $$ \left[\!\!\begin{array}{cc} 2 & -P_{11}-1 \\ -P_{11}-1 & P_{11}+2 \end{array}\!\!\right]\succeq 0 $$ if and only if $$ -\sqrt{3}\le P_{11}\le \sqrt{3}. $$ Therefore, choosing P11 = −1, we have that $$ P=W^{\ast}P_{F}W= \left[\!\!\begin{array}{cc} 1 & -1\\ 1 & 0 \end{array}\!\!\right] \left[\!\!\begin{array}{cc} -1 & 0\\\ 0 & 0 \end{array}\!\!\right] \left[\!\!\begin{array}{cc} 1 & 1\\ -1 & 0 \end{array}\!\!\right] = \left[\!\!\begin{array}{cc} -1 & -1\\ -1 & -1 \end{array}\!\!\right] $$ solves the KYP inequality (16). In particular, by Theorem 3.6 we obtain that for the Popov functions $$\Phi _{F}(z)\in \mathbb{K}(z)$$ and $$\Phi (z)\in \mathbb{K}(z)$$ we have $$\Phi _{F}\left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ and $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$. Remark 3.8 The result of Theorem 3.6 is analogous to the continuous-time result in the study by Reis et al. (2015). To see this, replace positivity of the Popov function on the unit circle by positivity on the imaginary axis in (a) and replace $$\mathcal M(P)$$ by its continuous-time analogue. However, in the study by Reis et al. (2015) the assumption of R-controllability was alternatively replaced by the condition that the Popov function has full rank and $$(E, A, B)$$ is sign-controllable. To adapt this to the discrete-time setting we would need a discrete-time analogue of Clements et al. (1997, Theorem 6.1), which provides the characterizations via sign-controllability in the ODE case. 4. Structure of palindromic matrix pencils In this section we are concerned with palindromic matrix pencils $$z\mathcal{A}^{\ast } - \mathcal{A} \in \mathbb{K}[z]^{n\times n}$$. For the investigation of these palindromic matrix pencils we first introduce so-called quasi-Hermitian matrices. Then, we show characterizations of the inertia of palindromic matrix pencils similar to what was done in the studies by Clements & Glover (1989), Clements et al. (1997), Reis (2011) and Voigt (2015), in the case of so-called even matrix pencils. The concept of quasi-Hermitian matrices is an extension to the notion of Hermitian and skew-Hermitian matrices. These are matrices $$\mathcal{A}\in \mathbb{K}^{n \times n}$$ with the property $$\mathcal{A}=\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast }$$ for some ω ∈ [0, 2π). They have the special property that every eigenvalue lies on the line with angle ω/2 through the origin. We can extend the notion of inertia for Hermitian matrices to quasi-Hermitian matrices. Definition 4.1 Let $$\mathcal{A}=\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast }$$ be quasi-Hermitian with ω ∈ [0, 2π). Then, the inertia of $$\mathcal{A}$$ along $$\frac{\omega }{2}$$ is $$ \operatorname{In}(\mathcal{A}) := \operatorname{In}_{\frac{\omega}{2}}(\mathcal{A}):=(n_{+},\,n_{0},\,n_{-}), $$ where n+, n0 and n− denote the number of eigenvalues $$\lambda =r\mathrm{e}^{\mathrm{i}\frac{\omega }{2}}$$ where r is positive, zero or negative, respectively. We omit the subscript $$\frac{\omega }{2}$$ in $$ \operatorname{In}_{\frac{\omega }{2}}(\mathcal{A})$$ if the angle is clear from the context. Similar to the Hermitian case, also in the quasi-Hermitian case we have a canonical form under congruence transformations. Theorem 4.2 The inertia of a quasi-Hermitian matrix is invariant under congruence transformations. On the other hand, if $$\mathcal{A},\,\mathcal{B}\in \mathbb{K}^{n \times n}$$ are two quasi-Hermitian matrices having the same inertia with respect to the same ω ∈ [0, 2π), then there exists some invertible $$U\in \mathbb{K}^{n \times n}$$ such that $$ U^{\ast}\mathcal{A} U=\mathcal{B}, $$ i.e. $$\mathcal{A}$$ and $$\mathcal{B}$$ are congruent. Proof. See the article by Ikramov (2001). We are now interested in a structure-preserving canonical form revealing the eigenstructure of a palindromic matrix pencil. Theorem 4.3 (Palindromic Kronecker canonical form) (Schröder, 2008) Let $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$ be a palindromic matrix pencil. Then, there exists some invertible $$U\in \mathbb{C}^{n\times n}$$ such that \begin{align} U^{\ast}\left(z\mathcal{A}^{\ast}-\mathcal{A}\right)U = \operatorname{diag}\left({D_{1}(z),\ldots,D_{l}(z)} \right) \end{align} (21) for some $$l\in \mathbb{N}$$ is in palindromic Kronecker canonical form (PKCF), where each block $$D_{j}(z)\in \mathbb{C}[z]^{k_{j}\times k_{j}}$$, $$k_{j}\in \mathbb{N}$$, is of one of the forms shown in Table 3 and The PKCF is unique up to permutations of the blocks and the quantities εj ∈{−1, 1} are called sign-characteristics. A closely related version of the above theorem was developed in the article by Horn & Sergeichuk (2006). Remark 4.4 We have multiplied the sign-characteristics of the blocks of type P4 occurring in the thesis by Schröder (2008) with −1 in order to simplify some of the upcoming results. This is justified by the fact that if $$\tilde{D}_{j}(z)$$ with sign-characteristic $$\tilde{\varepsilon }_{j}$$ corresponds to a block of type P4 introduced in the thesis by Schröder (2008), then $$D_{j}(z)=-U^{\ast }\tilde{D}_{j}(z)U$$ with $$ U=\mathrm{i} \left[\begin{array}{@{}cc@{}} I_{k_{j}/2} & \\ & -I_{k_{j}/2} \end{array}\right]\in\mathbb{C}^{k_{j}\times k_{j}} $$ is a block of type P4 with sign-characteristic $$\varepsilon _{j}=-\tilde{\varepsilon }_{j}$$ according to Theorem 4.3. Remark 4.5 By analysing the eigenstructure of the blocks in the form (21) we obtain (a) Blocks of type P1 correspond to eigenvalues λ and $${1}/{\overline{\lambda }}$$ with $$|\lambda |\neq 1$$, i.e. these eigenvalues occur in pairs $$\left \{\lambda ,{1}/{\overline{\lambda }}\right \}$$. In particular, this holds for the pairing $$\{0,\infty \}$$. (b) Blocks of type P2, P3 and P4 correspond to eigenvalues λ with $$\vert\lambda\vert=1 $$. (c) Blocks of type P5 correspond to rank deficiency of the pencil, i.e. they correspond to singular blocks. Consider the palindromic matrix pencil $$\mathcal{P}(z)=z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$. By inserting $$\mathrm{e}^{\mathrm{i}\omega} $$ for the polynomial variable z we obtain $$ \mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right)=\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast}-\mathcal{A} = \mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\left(\mathrm{i} \mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\mathcal{A}-\mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\mathcal{A}^{\ast}\right) $$ and hence $$ \mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right)^{\ast}= -\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\left(\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\mathcal{A}-\mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\mathcal{A}^{\ast}\right)= \left(-\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\right)^{2}\mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right)=-\mathrm{e}^{-\mathrm{i}\omega} \mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right). $$ Thus, $$\mathcal{P}\left (\mathrm{e}^{\mathrm{i}\omega} \right )$$ is quasi-Hermitian and has well-defined inertia. Investigating the block structure of the PKCF leads to the following result: Lemma 4.6 Assume that $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$ is in PKCF, i.e. it holds that $$z\mathcal{A}^{\ast }-\mathcal{A} = \operatorname{diag}\left ({D_{1}(z),\ldots ,D_{l}(z)} \right )$$ for some $$l\in \mathbb{N}$$. Then, the inertia pattern of each block $$D_{j}(z)\in \mathbb{C}[z]^{k_{j}\times k_{j}}$$, $$k_{j}\in \mathbb{N}$$, is given as in Table 3. Proof. See the thesis by Bankmann (2016, Lemma 4.11). Remark 4.7 The results from Lemma 4.6 can be used to determine the block structure of a pencil $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$ in the form (21), given the inertia patterns for ω ∈ [0, 2π). Note that blocks of type P1 have a very simple inertia pattern and thus from a general pattern $$ \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}-\mathcal{A}\right) = \left(k_{1},k_{2},k_{3}\right) $$ —except for the case where k2 = 0, i.e. all blocks are of type P1—we cannot tell whether or how many blocks of type P1 are present in the PKCF. Table 3. Blocks D(z) occurring in palindromic Kronecker canonical form and their inertia $$ \operatorname{In}\left ( D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ for ω ∈ [0, 2π), where the addition of two inertia tuples has to be understood component-wise Type Dimension D(z) Parameters $$ \operatorname{In}\left (D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ P1 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda )\\\hline zJ_{k}(\overline{\lambda} )-F_{k} &\\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon \!\in\! \{-1,1\},\\ \lambda \!=\!r\mathrm{e}^{\mathrm{i}\theta },\,{|r|<1,}\\{\theta \!\in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c} \left (k,0,k\right )\\[23pt]\end{array}$$ P2 2k + 1 $$\varepsilon \left [ \begin{array}{c|c|c} & & zF_{k}-J_{k}(\lambda )\\\hline & z\mathrm{e}^{-\mathrm{i}\frac{\theta }{2}}- \mathrm{e}^{\mathrm{i}\frac{\theta }{2}} & -{\left (e^{ k}_{1}\right )}^{T}\\\hline zJ_{k}(\overline \lambda )-F_{k} & ze^{ k}_{1} \\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon\! \in\! \{-\!1,1\},\\ \lambda \!=\!\mathrm{e}^{\mathrm{i}\theta }\!,{\theta\! \in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c}\left (k,0,k\right )\!+\! \operatorname{In}\!\left (\varepsilon (\omega \!-\!\theta )\right )\\[11pt]\end{array}$$ P3 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda ) \\\hline zJ_{k}(\overline \lambda )-F_{k} & (z-1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right] $$ $$\begin{array}{l}{k\in \mathbb{N}_{0},}\,\varepsilon \in \{-1,1\},\\ \lambda =\mathrm{e}^{\mathrm{i}\theta }\!,{\theta \in [0,2\pi )}\end{array}$$ $$\begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq \theta \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = \theta \end{cases} $$ P4 2k $$\varepsilon \left [ \begin{array}{c|c} & zF_{k}-J_{k}(1)\\\hline zJ_{k}(1)-F_{k} & \textrm{i}(z+1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right ]$$ $$\ k\in \mathbb{N}_{0},\,\!\varepsilon \!\in\! \{-1,1\}$$ $$ \begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq 0 \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = 0 \end{cases} $$ P5 2k + 1 $$\left [ \begin{array}{c|c} & zS_{R} - {S_{L}^{T}} \\\hline zS_{L} - {S_{R}^{T}} & \\ \end{array}\right ] $$ $$\ k\in \mathbb{N}_{0}$$ $$\ \left (k,1,k\right )$$ Type Dimension D(z) Parameters $$ \operatorname{In}\left (D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ P1 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda )\\\hline zJ_{k}(\overline{\lambda} )-F_{k} &\\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon \!\in\! \{-1,1\},\\ \lambda \!=\!r\mathrm{e}^{\mathrm{i}\theta },\,{|r|<1,}\\{\theta \!\in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c} \left (k,0,k\right )\\[23pt]\end{array}$$ P2 2k + 1 $$\varepsilon \left [ \begin{array}{c|c|c} & & zF_{k}-J_{k}(\lambda )\\\hline & z\mathrm{e}^{-\mathrm{i}\frac{\theta }{2}}- \mathrm{e}^{\mathrm{i}\frac{\theta }{2}} & -{\left (e^{ k}_{1}\right )}^{T}\\\hline zJ_{k}(\overline \lambda )-F_{k} & ze^{ k}_{1} \\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon\! \in\! \{-\!1,1\},\\ \lambda \!=\!\mathrm{e}^{\mathrm{i}\theta }\!,{\theta\! \in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c}\left (k,0,k\right )\!+\! \operatorname{In}\!\left (\varepsilon (\omega \!-\!\theta )\right )\\[11pt]\end{array}$$ P3 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda ) \\\hline zJ_{k}(\overline \lambda )-F_{k} & (z-1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right] $$ $$\begin{array}{l}{k\in \mathbb{N}_{0},}\,\varepsilon \in \{-1,1\},\\ \lambda =\mathrm{e}^{\mathrm{i}\theta }\!,{\theta \in [0,2\pi )}\end{array}$$ $$\begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq \theta \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = \theta \end{cases} $$ P4 2k $$\varepsilon \left [ \begin{array}{c|c} & zF_{k}-J_{k}(1)\\\hline zJ_{k}(1)-F_{k} & \textrm{i}(z+1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right ]$$ $$\ k\in \mathbb{N}_{0},\,\!\varepsilon \!\in\! \{-1,1\}$$ $$ \begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq 0 \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = 0 \end{cases} $$ P5 2k + 1 $$\left [ \begin{array}{c|c} & zS_{R} - {S_{L}^{T}} \\\hline zS_{L} - {S_{R}^{T}} & \\ \end{array}\right ] $$ $$\ k\in \mathbb{N}_{0}$$ $$\ \left (k,1,k\right )$$ View Large Table 3. Blocks D(z) occurring in palindromic Kronecker canonical form and their inertia $$ \operatorname{In}\left ( D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ for ω ∈ [0, 2π), where the addition of two inertia tuples has to be understood component-wise Type Dimension D(z) Parameters $$ \operatorname{In}\left (D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ P1 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda )\\\hline zJ_{k}(\overline{\lambda} )-F_{k} &\\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon \!\in\! \{-1,1\},\\ \lambda \!=\!r\mathrm{e}^{\mathrm{i}\theta },\,{|r|<1,}\\{\theta \!\in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c} \left (k,0,k\right )\\[23pt]\end{array}$$ P2 2k + 1 $$\varepsilon \left [ \begin{array}{c|c|c} & & zF_{k}-J_{k}(\lambda )\\\hline & z\mathrm{e}^{-\mathrm{i}\frac{\theta }{2}}- \mathrm{e}^{\mathrm{i}\frac{\theta }{2}} & -{\left (e^{ k}_{1}\right )}^{T}\\\hline zJ_{k}(\overline \lambda )-F_{k} & ze^{ k}_{1} \\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon\! \in\! \{-\!1,1\},\\ \lambda \!=\!\mathrm{e}^{\mathrm{i}\theta }\!,{\theta\! \in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c}\left (k,0,k\right )\!+\! \operatorname{In}\!\left (\varepsilon (\omega \!-\!\theta )\right )\\[11pt]\end{array}$$ P3 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda ) \\\hline zJ_{k}(\overline \lambda )-F_{k} & (z-1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right] $$ $$\begin{array}{l}{k\in \mathbb{N}_{0},}\,\varepsilon \in \{-1,1\},\\ \lambda =\mathrm{e}^{\mathrm{i}\theta }\!,{\theta \in [0,2\pi )}\end{array}$$ $$\begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq \theta \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = \theta \end{cases} $$ P4 2k $$\varepsilon \left [ \begin{array}{c|c} & zF_{k}-J_{k}(1)\\\hline zJ_{k}(1)-F_{k} & \textrm{i}(z+1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right ]$$ $$\ k\in \mathbb{N}_{0},\,\!\varepsilon \!\in\! \{-1,1\}$$ $$ \begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq 0 \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = 0 \end{cases} $$ P5 2k + 1 $$\left [ \begin{array}{c|c} & zS_{R} - {S_{L}^{T}} \\\hline zS_{L} - {S_{R}^{T}} & \\ \end{array}\right ] $$ $$\ k\in \mathbb{N}_{0}$$ $$\ \left (k,1,k\right )$$ Type Dimension D(z) Parameters $$ \operatorname{In}\left (D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ P1 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda )\\\hline zJ_{k}(\overline{\lambda} )-F_{k} &\\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon \!\in\! \{-1,1\},\\ \lambda \!=\!r\mathrm{e}^{\mathrm{i}\theta },\,{|r|<1,}\\{\theta \!\in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c} \left (k,0,k\right )\\[23pt]\end{array}$$ P2 2k + 1 $$\varepsilon \left [ \begin{array}{c|c|c} & & zF_{k}-J_{k}(\lambda )\\\hline & z\mathrm{e}^{-\mathrm{i}\frac{\theta }{2}}- \mathrm{e}^{\mathrm{i}\frac{\theta }{2}} & -{\left (e^{ k}_{1}\right )}^{T}\\\hline zJ_{k}(\overline \lambda )-F_{k} & ze^{ k}_{1} \\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon\! \in\! \{-\!1,1\},\\ \lambda \!=\!\mathrm{e}^{\mathrm{i}\theta }\!,{\theta\! \in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c}\left (k,0,k\right )\!+\! \operatorname{In}\!\left (\varepsilon (\omega \!-\!\theta )\right )\\[11pt]\end{array}$$ P3 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda ) \\\hline zJ_{k}(\overline \lambda )-F_{k} & (z-1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right] $$ $$\begin{array}{l}{k\in \mathbb{N}_{0},}\,\varepsilon \in \{-1,1\},\\ \lambda =\mathrm{e}^{\mathrm{i}\theta }\!,{\theta \in [0,2\pi )}\end{array}$$ $$\begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq \theta \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = \theta \end{cases} $$ P4 2k $$\varepsilon \left [ \begin{array}{c|c} & zF_{k}-J_{k}(1)\\\hline zJ_{k}(1)-F_{k} & \textrm{i}(z+1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right ]$$ $$\ k\in \mathbb{N}_{0},\,\!\varepsilon \!\in\! \{-1,1\}$$ $$ \begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq 0 \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = 0 \end{cases} $$ P5 2k + 1 $$\left [ \begin{array}{c|c} & zS_{R} - {S_{L}^{T}} \\\hline zS_{L} - {S_{R}^{T}} & \\ \end{array}\right ] $$ $$\ k\in \mathbb{N}_{0}$$ $$\ \left (k,1,k\right )$$ View Large 5. Inertia of palindromic pencils in optimal control Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given. We consider palindromic matrix pencils arising in the optimal control problem as in (14) of the form $$ z\mathcal{A}^{\ast}-\mathcal{A} = \left[\begin{array}{@{}ccc@{}} 0 & zE-A & -B \\ zA^{\ast}-E^{\ast} & (z-1)Q & (z-1)S \\ zB^{\ast} & (z-1)S^{\ast} & (z-1)R \end{array}\right]\in \mathbb{K}[z]^{(2n+m)\times (2n+m)}. $$ If we insert $$\mathrm{e}^{\mathrm{i}\omega} $$ into (14) for z we obtain the quasi-Hermitian matrix \begin{align} \mathcal{D}(\omega):= \mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}} \left(\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\mathcal{A}-\mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\mathcal{A}^{\ast}\right) = \mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}} \left[\begin{array}{@{}ccc@{}} 0 & E_{\omega}-A_{\omega} & B_{\omega} \\ E_{\omega}^{\ast}-A_{\omega}^{\ast} & Q_{\omega} & S_{\omega} \\ B_{\omega}^{\ast} & S_{\omega}^{\ast} & R_{\omega} \end{array}\right]\in\mathbb{C}^{(2n+m)\times (2n+m)} \end{align} (22) with $$E_{\omega }=-\textrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega }{2}}E$$, $$A_{\omega }=-\textrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega }{2}}A$$, $$B_{\omega }=\textrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega }{2}}B$$, Qω = sωQ, Sω = sωS and Rω = sωR, where $$s_{\omega }= \textrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega }{2}}-\textrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega }{2}}=2 \sin \left (\frac{\omega }{2}\right )$$. Lemma 5.1 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ and consider the matrix $$\mathcal{D}(\omega )$$ as in (22) with ω such that $$\det \left (E_{\omega }-A_{\omega }\right )\neq 0$$. Furthermore, let $$ U= \left[\begin{array}{@{}ccc@{}} I_{n} & 0 & \left(E_{\omega}^{\ast}-A_{\omega}^{\ast}\right)^{-1}\left(Q_{\omega}\left(E_{\omega}-A_{\omega}\right)^{-1}B_{\omega} - S_{\omega}\right) \\ 0 & I_{n} & -\left(E_{\omega}-A_{\omega}\right)^{-1}B_{\omega} \\ 0 & 0 & I_{m} \end{array}\right]\in\mathbb{C}^{(2n+m)\times (2n+m)}. $$ Then, $$\mathcal{D}(\omega )$$ is congruent to $$ U^{\ast}\mathcal{D}(\omega)U= \mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}} \left[\begin{array}{@{}ccc@{}} 0 & E_{\omega}-A_{\omega} & 0 \\ E_{\omega}^{\ast}-A_{\omega}^{\ast} & Q_{\omega} & 0 \\ 0 & 0 & 2\sin\left(\frac{\omega}{2}\right)\Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right) \end{array}\right]. $$ Proof. See the thesis by Bankmann (2016, Lemma 4.13). Theorem 5.2 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given with corresponding Popov function $$\Phi (z)\in \mathbb{K}[z]^{m\times m}$$ and $$\textrm{rk}_{\mathbb{K}(z)} \Phi (z)=q$$ for some $$q\in \mathbb{N}_{0}$$. Assume that $$(E, A, B)$$ has no uncontrollable modes on the unit circle. Then, the following are equivalent: (a) The Popov function Φ(z) is positive semi-definite on the unit circle, i.e. $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π). (b) The following conditions for the PKCF of $$z\mathcal{A}^{\ast }-\mathcal{A}$$ as in (21) hold: (i) There are no blocks of type P2 corresponding to eigenvalues $$\lambda =\mathrm{e}^{\mathrm{i}\theta },\, \theta \neq 0$$, and all blocks of type P3 have negative sign-characteristic. (ii) The number of blocks of type P2 corresponding to an eigenvalue λ = 1 with positive sign-characteristic is greater by q than the number of those with negative sign-characteristic. (c) The following conditions for the PKCF of $$z\mathcal{A}^{\ast }-\mathcal{A}$$ as in (21) hold: (i’) There are no blocks of type P2 corresponding to eigenvalues $$\lambda =\mathrm{e}^{\mathrm{i}\theta },\, \theta \neq 0$$. (ii’) The number of blocks of type P2 corresponding to an eigenvalue λ = 1 with positive sign-characteristic is greater by q than the number of those with negative sign-characteristic. Proof. The strategy of the proof is similar to the one in the thesis by Voigt (2015, Theorem 3.4.2) for the continuous-time case. First note that since $$(E, A, B)$$ has no uncontrollable modes on the unit circle we can find a feedback matrix $$F\in \mathbb{K}^{m\times n}$$ such that $$zE - (A+BF) \in \mathbb{K}[z]^{n\times n}$$ has no eigenvalues on the unit circle. Then, by Lemma 3.3 and the fact that the palindromic pencil $$z\mathcal{A}_{F}^{\ast }-\mathcal{A}_{F}$$ corresponding to $$(E,A+BF,B)$$ is connected to $$z\mathcal{A}^{\ast }-\mathcal{A}$$ via $$\mathcal{A}_{F}= U_{F}^{\ast }\mathcal{A} U_{F}$$, where $$ U_{F}:= \left[\begin{array}{@{}ccc@{}} I_{n} & 0 & 0 \\ 0 & I_{n} & 0 \\ 0 & F & I_{m} \end{array}\right]\in\mathbb{K}^{(2n+m)\times (2n+m)}, $$ we can assume without loss of generality that (E, A) has no eigenvalues on the unit circle. Now we show that (a) implies (b). Therefore, assume that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π). Then, in particular we have $$ \operatorname{In}\left(\Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right)\right)=\left(q-a(\omega),m-q+a(\omega),0\right) $$ for all ω ∈ (0, 2π), where $$a:(0,2\pi )\to \mathbb{N}_{0}$$ is some function which is zero for almost all ω ∈ (0, 2π). Hence, by Lemma 5.1 and Theorem 4.2 we obtain $$ \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}-\mathcal{A} \right) = \left( n + q - a(\omega), m-q+a(\omega), n \right) $$ for ω ∈ (0, 2π). Again, by Theorem 4.2 the inertia of $$\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast }-\mathcal{A}$$ coincides with the inertia of the PKCF of $$z\mathcal{A}^{\ast }-\mathcal{A}$$ as in (21) evaluated at $$\mathrm{e}^{\mathrm{i}\omega} $$. Since by Theorem 4.3 the block structure of the PKCF is uniquely determined, we can proceed by identifying blocks by their inertia patterns. Note that $$\textrm{rk}_{\mathbb{K}(z)} \left (z\mathcal{A}^{\ast }-\mathcal{A}\right ) = 2n + q$$, since $$z\mathcal{A}^{\ast }-\mathcal{A}$$ can only have a finite amount of rank drops and due to Lemma 5.1 and the regularity of zE − A there exist infinitely many values $$\lambda \in \mathbb{C}$$ for which $$ \operatorname{rk}(\lambda \mathcal{A}^{\ast }-\mathcal{A})=2n + q$$. From Lemma 4.6 we can infer that we have exactly 2n + m − (2n + q) = m − q blocks of type P5 in the PKCF of $$z\mathcal{A}^{\ast }-\mathcal{A}$$, since these are the only rank-deficient blocks. Thus, since $$ \operatorname{rk} (\mathcal{A}^{\ast }-\mathcal{A}) = 2r$$, where $$r:= \operatorname{rk} \left [E-A\enspace B\right ]$$, the number of blocks of type P2 or P4 corresponding to an eigenvalue λ = 1 is exactly 2(n − r) + q. Then, removing the blocks of type P5 from the PKCF of $$z\mathcal{A}^{\ast }-\mathcal{A}$$ yields a matrix pencil $$z\mathcal{A}^{\ast }_{1}-\mathcal{A}_{1}\in \mathbb{K}[z]^{(2n_{1}+q)\times (2n_{1}+q)}$$ in PKCF with full normal rank and inertia $$ \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}_{1}-\mathcal{A}_{1}\right)=\left( n_{1} + q - a(\omega), a(\omega), n_{1} \right) $$ on (0, 2π). Then, by Lemma 4.6, there are q blocks of type P2 with corresponding eigenvalue λ = 1 and positive sign-characteristic, since these are present in every combination of blocks with an inertia pattern of the form (k + 1, 0, k) independent of ω > 0 . Removing these blocks leads to the pencil $$z\mathcal{A}^{\ast }_{2}-\mathcal{A}_{2}\in \mathbb{K}[z]^{2n_{2}\times 2n_{2}}$$ in PKCF with inertia $$ \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}_{2}-\mathcal{A}_{2}\right)=\left( n_{2} - a(\omega), a(\omega), n_{2} \right) $$ on (0, 2π). Furthermore, from Lemma 4.6 we deduce that there are no blocks of type P2 corresponding to eigenvalues $$\lambda =\mathrm{e}^{\mathrm{i}\theta },\, \theta \neq 0$$. Thus, all blocks of type P3 have negative sign-characteristic, since these are the only blocks with an inertia pattern of the form (k − 1, 1, k) for exactly one value of ω > 0. This shows statement (i). Removing these blocks, we obtain a matrix pencil $$z\mathcal{A}^{\ast }_{3}-\mathcal{A}_{3}\in \mathbb{K}[z]^{2n_{3}\times 2n_{3}}$$ in PKCF with inertia $$ \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}_{3}-\mathcal{A}_{3}\right)=\left( n_{3}, 0, n_{3} \right) $$ on (0, 2π). The inertia pattern of $$z\mathcal{A}_{3}^{\ast }-\mathcal{A}_{3}$$ together with Lemma 4.6 reveals that the remaining blocks of type P2 corresponding to an eigenvalue λ = 1 are split up equally into those with positive and those with negative sign-characteristic. This shows (ii) and thus statement (b). The proof that (c) follows from (b) is clear, since condition (i’) follows immediately from condition (i) and conditions (ii) and (ii’) coincide. Now let the conditions (i’) and (ii’) hold. Again, by Lemma 5.1 and Theorem 4.2, for ω ∈ (0, 2π) we obtain \begin{align*} \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}-\mathcal{A} \right) =& \left( n, 0, n \right) + \operatorname{In}\left(\Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right)\right)\\ =& \left( n + m_{1} - a_{1}(\omega), m-m_{1}-m_{2} + a_{1}(\omega) + a_{2}(\omega), n + m_{2} - a_{2}(\omega) \right) \end{align*} and functions $$a_{i}:(0,2\pi )\to \mathbb{N}_{0}$$, i = 1, 2, which are zero for almost all ω ∈ (0, 2π) such that m1 + m2 = q. We now have to show that m2 = 0 = a2(ω). Then, the blocks of type P2 with positive sign-characteristic are the only ones leading to an inertia pattern of the form (k + 1, 0, k) for ω > θ. The only blocks that could compensate the additional positive eigenvalue for ω > θ are blocks of type P2 with negative sign-characteristic. By condition (i’) we are only allowed to take such blocks with θ = 0. By condition (ii’) then we obtain that n + m1 = (n + m2) + q and thus m2 = 0, m1 = q. Hence, we have $$ \operatorname{In}\left(\Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right)\right)=\left( q - a_{1}(\omega), m-q + a_{1}(\omega) + a_{2}(\omega), - a_{2}(\omega) \right). $$ Since the inertia of a quasi-Hermitian matrix by definition is a triple of non-negative integers, this implies a2 ≡ 0 and thus $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ (0, 2π). Then, by continuity, we also have that Φ(1) $$\succeq $$ 0. Example 5.3 (Example 2.7 revisited) We consider the system $$(E, A, B, Q, S, R)$$ with corresponding system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form as in (4), (5), and Example 3.7. The associated palindromic pencil $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{5\times 5}$$ as in (14) is given by \begin{align} z\mathcal{A}^{\ast}-\mathcal{A}=\left[ \begin{array}{@{}cc|cc|c@{}} 0 & 0 & 1 & -1 & -1\\ 0 & 0 & -1 & z &0 \\ \hline -z & z & z-1 & 0 &0\\ z&-1 & 0 & z-1 &0\\\hline z & 0 & 0 & 0 &z-1 \end{array}\right]. \end{align} (23) Transforming the matrix $$\mathcal{A}$$ to the corresponding matrix $$\mathcal{A}_{F}$$ of the system in feedback equivalence form (5) via $$ U_{F} := \left[\begin{array}{@{}ccc@{}} W^{\ast} & 0 & 0 \\ 0 & T & 0 \\ 0 & FT & I_{m} \end{array}\right]= \left[ \begin{array}{@{}cc|cc|c@{}} 1 & -1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & 1 \end{array}\right]\in\mathbb{K}^{5\times 5} $$ we obtain $$ \mathcal{A}_{F}= U_{F}^{\ast} \mathcal{A} U_{F}= \left[ \begin{array}{@{}cc|cc|c@{}} 0 & 0 & 1 & 0 & \phantom{-}1\\ 0 & 0 & 0 & 1 &-1\\\hline 1 & 0 & 2 & 1 &0\\ 0 & 0 & 1 & 1 &0\\\hline 0 & 0 & 0 & 0 &1 \end{array}\right]. $$ The matrix $$\mathcal{A}_{F}$$ can be further transformed to \begin{align} U^{\ast}\left(z\mathcal{A}_{F}^{\ast}-\mathcal{A}_{F}\right)U= \left[ \begin{array}{@{}cc|cc|c@{}} 0 & -1 & 0 & 0 & 0 \\ z & 0 & 0 & 0 & 0 \\\hline 0 & 0 & 0 & z-(2+\sqrt{3}) & 0 \\ 0 & 0 & (2+\sqrt{3})z-1 & 0 & 0 \\\hline 0 & 0 & 0 & 0 & z-1 \end{array} \right] \end{align} (24) in PKCF as in (21) via $$ U= \left[ \begin{array}{@{}ccccc@{}} 0 & -1 & -1+\sqrt{3} & -\frac{3}{2}-\sqrt{3} & 0 \\ 1 & -1 & -1+\frac{1}{\sqrt{3}} & -1-\frac{\sqrt{3}}{2} & 0 \\ 0 & 0 & -\frac{1}{\sqrt{3}} & \frac{1}{4} \left(1+\sqrt{3}\right) & -\frac{1}{\sqrt{2}} \\ 0 & 1 & 1-\frac{1}{\sqrt{3}} & \frac{1}{2} \left(2+\sqrt{3}\right) & 0 \\ 0 & 0 & -1+\frac{1}{\sqrt{3}} & -1-\frac{\sqrt{3}}{2} & 0 \end{array} \right]. $$ From (24) we see that the PKCF of $$z\mathcal{A}_{F}^{\ast }-\mathcal{A}_{F}$$ and thus also of $$z\mathcal{A}^{\ast }-\mathcal{A}$$ consists of a 2 × 2 block of type P1 corresponding to the eigenvalues $$\{0,\infty \}$$, a 2 × 2 block of type P1 corresponding to the eigenvalues $$\left \{2+\sqrt{3},2-\sqrt{3}\right \}$$ and a 1 × 1 block of type P2 corresponding to the eigenvalue one. Furthermore, for the Popov function ΦF(z) it holds that $$\textrm{rk}_{\mathbb{K}(z)} \Phi _{F}(z)=1$$. Thus, we have shown that the assumptions of Theorem 5.2(b) are fulfilled and hence $$\Phi _{F}\left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π). Thus, we have confirmed the result obtained in Example 3.7. 6. Lur’e equations In this section we characterize solvability of Lur’e equations for explicit as well as for IDEs in a similar way as in the article by Reis et al. (2015) for continuous-time systems. Finding a solution of the Lur’e equation means finding $$X=X^{\ast }\in \mathbb{K}^{n \times n}$$, $$K\in \mathbb{K}^{q\times n}$$ and $$L\in \mathbb{K}^{q\times m}$$ such that \begin{align} \mathcal M (X) = \left[\begin{array}{@{}cc@{}} A^{\ast}XA-E^{\ast}XE + Q & A^{\ast}XB + S \\ B^{\ast}XA + S^{\ast} & B^{\ast}XB + R \end{array}\right] ={\!}_{\mathcal{V}_{(E,A,B)}} \left[\begin{array}{@{}c@{}} K^{\ast}\\ L^{\ast} \end{array}\right] \left[\begin{array}{@{}cc@{}} K & L \end{array}\right], \end{align} (25) where $$q:=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. If X is a solution of the KYP inequality (16), then we can always find $$K\in \mathbb{K}^{p\times n}$$ and $$L\in \mathbb{K}^{p\times m}$$ for some $$p\in \mathbb{N}_{0}$$ such that (25) holds. The next result shows that for such solutions it holds that $$p\geq q $$. Thus, in other words, we are interested in the existence of solutions of (25) with minimal rank q. Proposition 6.1 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. Further, let $$ (X,\,K,\,L)\in\mathbb{K}^{n \times n}\times\mathbb{K}^{q\times n}\times\mathbb{K}^{q\times m} $$ be a solution of the Lur’e equation (25) and assume that for $$M\in \mathbb{K}^{p\times n}$$ and $$N\in \mathbb{K}^{p\times m}$$ also the triple (X, M, N) fulfils (25). Then, we have q ≤ p and \begin{align} \textrm{rk}_{\mathbb{K}(z)} \left[\begin{array}{@{}cc@{}} zE-A & -B \\ (z-1)K &(z-1)L \end{array}\right] = n+q. \end{align} (26) Proof. See the thesis by Bankmann (2016, Proposition 5.1). Note that in the continuous-time case (Reis et al., 2015) instead of rank minimality the condition (26) was used to define solutions for the Lur’e equation (25). Proposition 6.1 shows that both versions are indeed equivalent. In the following we will derive certain deflating subspaces of BVD and palindromic matrix pencils, respectively, from a solution of the Lur’e equation (25). First, we do this for the case of EDEs. Afterwards, based on these results, we do the generalization to the implicit case with the help of feedback transformations similarly to the approach in Section 3. 6.1. Explicit difference equations In the EDE case, i.e. systems $$(I_{n},A,B,Q,S,R) \in \Sigma^w_{m,n}(\mathbb{K})$$ finding a solution of the Lur’e equation (25) reduces to: For $$q:=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$ find $$X\in \mathbb{K}^{n \times n}$$, $$K\in \mathbb{K}^{q\times n}$$ and $$L\in \mathbb{K}^{q\times m}$$ such that \begin{align} \mathcal M (X)= \left[\begin{array}{@{}cc@{}} A^{\ast}XA-X + Q & A^{\ast}XB + S \\ B^{\ast}XA + S^{\ast} & B^{\ast}XB + R \end{array}\right]= \left[\begin{array}{@{}c@{}} K^{\ast}\\ L^{\ast} \end{array}\right] \left[\begin{array}{@{}cc@{}} K & L \end{array}\right]\!. \end{align} (27) The next result is an analogous version of Reis (2011, Lemma 12) for the discrete-time case. Lemma 6.2 Let $$(I_{n},A,B,Q,S,R) \in \Sigma^w_{m,n}(\mathbb{K})$$ be given and let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. Furthermore, let $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π) such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} I_{n}-A\right )\neq 0$$ and let a Hermitian $$X\in \mathbb{K}^{n \times n}$$ be given with $$ \operatorname{rk} \mathcal M (X) =q. $$ Then, (27) has a solution $$(X,\,K,\,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$. Proof. The proof is analogous to the continuous-time case (Reis, 2011) and can be found in the thesis by Bankmann (2016, Lemma 5.2). Example 6.3 (Example 2.7 revisited) Consider the system $$(E, A, B, Q, S, R)$$ as in (4) and Example 3.7. We have seen in Example 3.7 that with $$ \mathcal M_{s} (P_{s})= \left[\begin{array}{@{}cc@{}} 2 & -P_{s}-1 \\ -P_{s}-1 & P_{s}+2 \end{array}\right] $$$$P_{s}= \sqrt{3}$$ solves the KYP inequality (16) for the EDE system $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)$$ as in (20). In particular, we have that $$ \operatorname{rk} \mathcal M_{s} (P_{s})= 1 = \textrm{rk}_{\mathbb{K}(z)} \Phi _{s}(z)$$ for the Popov function $$\Phi _{s}(z)\in \mathbb{K}(z)$$ of the EDE system. Thus, we obtain $$ \mathcal M_{s} (P_{s}) = \left[\begin{array}{@{}cc@{}} {\sqrt2} & 0\\ - \frac{\sqrt3+1}{\sqrt2} & 1 \end{array}\right] \left[\begin{array}{@{}cc@{}} 1 & 0\\ 0 & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} {\sqrt2} & - \frac{\sqrt3+1}{\sqrt2}\\ 0 & 1 \end{array}\right] $$ and hence, $$ (P_{s},\, K_{s},\, L_{s}) = \left(\sqrt 3,\,{\sqrt2},\, -\frac{\sqrt3+1}{\sqrt2}\right) $$ is a solution of the Lur’e equation (27). Now we are ready to show that the existence of a solution of the Lur’e equation (27) is equivalent to the existence of a certain deflating subspace of the palindromic matrix pencil as in (14). This result is the continuous-time analogue of Reis (2011, Theorem 11). Theorem 6.4 Let $$(I_{n},A,B,Q,S,R) \in \Sigma^w_{m,n}(\mathbb{K})$$ be given and consider the associated palindromic pencil $$z\mathcal{A}^{\ast }-\mathcal{A}$$ as in (14). Further, let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$ and assume that $$ \operatorname{rk}\,\left [\,I_{n}-A \enspace -\!B \,\right ] =n$$. Then, the following are equivalent: (a) There exists a solution $$(X,\,K,\,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \in \mathbb{K}^{q\times m}$$ of the Lur’e equation (27). (b) It holds that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π) such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} I_{n}-A\right )\neq 0$$. Furthermore, there exist matrices $$Y_{\mu },\,Y_{x}\in \mathbb{K}^{n\times (n+m)},\,Y_{u}\in \mathbb{K}^{m\times (n+m)}$$ and $$Z_{\mu },\,Z_{x}\in \mathbb{K}^{n\times (n+q)},\,Z_{u}\in \mathbb{K}^{m\times (n+q)}$$ such that for $$ Y=\left[\begin{array}{@{}c@{}} Y_{\mu}\\ Y_{x}\\ Y_{u} \end{array}\right],\qquad Z=\left[\begin{array}{@{}c@{}} Z_{\mu}\\ Z_{x}\\ Z_{u} \end{array}\right] $$ the following holds: (i) The matrix $$ \left[\begin{array}{@{}cc@{}} I_{n} -A & -B \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right] $$ has full row rank n. (ii) The space $$\mathcal Y= \operatorname{im} Y$$ is maximally $$\left (\mathcal{A}^{\ast }-\mathcal{A}\right )$$-neutral. (iii) There exist $$\tilde E,\,\tilde A \in \mathbb{K}^{(n+q)\times (n+m)}$$ such that $$\left (z\mathcal{A}^{\ast }-\mathcal{A}\right )Y=Z\left (z\tilde E - \tilde A\right )$$. Proof. Denote by $$C\in \mathbb{K}^{(n+m)\times n}$$ and $$C_{c}\in \mathbb{K}^{(n+m)\times m}$$ the right inverse and a basis matrix of the kernel of $$ \left[\begin{array}{@{}cc@{}} I_{n} -A & -B \end{array}\right], $$ respectively. Further let $$ \left[\begin{array}{@{}c@{}} C_{1}^{-} \\ C_{2}^{-} \end{array}\right] := \left[\begin{array}{@{}cc@{}} C_{11}^{-} &C_{12}^{-}\\ C_{21}^{-} &C_{22}^{-} \end{array}\right] := \left[\begin{array}{@{}cc@{}} C & C_{c} \end{array}\right]^{-1}, $$ where $$C_{1}^{-}=\left [\,I_{n}-A \quad -B \,\right ]\in \mathbb{K}^{n\times (n+m)}$$, $$C_{2}^{-}\in \mathbb{K}^{m\times (n+m)}$$, $$C_{11}^{-}=I_{n}-A\in \mathbb{K}^{n \times n}$$, $$C_{12}^{-}=-B\in \mathbb{K}^{n\times m}$$, $$C_{21}^{-}\in \mathbb{K}^{m\times n}$$ and $$C_{22}^{-}\in \mathbb{K}^{m\times m}$$. First assume that there exists a solution (X, K, L) of (27). Then, we have $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π) such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} I_{n}-A\right )\neq 0$$ . Set \begin{align} Y= \left[\begin{array}{@{}c@{}} Y_{\mu}\\ Y_{x}\\ Y_{u} \end{array}\right] = \left[\begin{array}{@{}cc@{}} X\left(A-I_{n}\right) & XB \\ I_{n} & 0 \\ 0 & I_{m} \end{array}\right] Z=\left[\begin{array}{@{}c@{}} Z_{\mu}\\ Z_{x}\\ Z_{u} \end{array}\right] = \left[\begin{array}{@{}cc@{}} I_{n} & 0 \\ \left(I_{n}-A^{\ast}\right)X & K^{\ast} \\ -B^{\ast}X & L^{\ast} \end{array}\right], \end{align} (28) and \begin{align} z\tilde E - \tilde A = \left[\begin{array}{@{}cc@{}} zI_{n}-A & -B \\ (z-1)K &(z-1)L \end{array}\right]. \end{align} (29) Property (i) follows, since $$ \operatorname{rk} \left[\begin{array}{cc} I_{n} -A & -B \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right] = \operatorname{rk} \left[\begin{array}{@{}cc@{}} I_{n} -A & -B \end{array}\right]=n $$ by assumption. For property (ii) we first note that for \begin{align} V:= \left[\begin{array}{@{}ccc@{}} I_{n} & 0 &0 \\ 0 & C_{11}^{-} &C_{12}^{-}\\ 0 & C_{21}^{-} &C_{22}^{-} \end{array}\right]\in\mathbb{K}^{(2n+m)\times (2n+m)} \end{align} (30) we have \begin{align} V^{-*}\left(\mathcal{A}^{\ast}-\mathcal{A}\right)V^{-1} = \left[\begin{array}{@{}ccc@{}} 0 & I_{n} & 0 \\ I_{n} & 0 & 0\\ 0 & 0 & 0 \end{array}\right] =:\mathcal{E}. \end{align} (31) Then, imY is maximally $$(\mathcal{A}^{\ast }-\mathcal{A})$$-neutral if and only if $$ \operatorname{im} \hat{Y}$$ is maximally $$\mathcal{E}$$-neutral, where $$ \hat{Y} := V Y \left[\begin{array}{cc} C & C_{c} \end{array}\right] = \left[\begin{array}{@{}cc@{}} -X & 0 \\ I_{n} & 0 \\ 0 & I_{m} \end{array}\right]. $$ On the one hand, $$ \operatorname{im} \hat{Y}$$ is $$\mathcal{E}$$-neutral, since $$ \hat{Y}^{\ast}\mathcal{E} \hat{Y} = \left[\begin{array}{@{}cc@{}} -X+X& 0\\ 0 & 0 \end{array}\right]= 0. $$ On the other hand, we have that $$n+m = \operatorname{rk} \hat{Y}$$ and the rank of every $$\mathcal{E}$$-neutral space is bounded from above by n + m. Therefore, $$ \operatorname{im} \hat{Y}$$ is maximally $$\mathcal{E}$$-neutral which shows (ii). Finally, we have (iii) by \begin{align*} \left(z\mathcal{A}^{\ast}-\mathcal{A}\right)Y =& \left[\begin{array}{@{}cc@{}} zI_{n} - A & -B \\ z \left(A^{\ast}X\left(A-I_{n}\right)+Q\right) - X\left(A-I_{n}\right)-Q & z\left(A^{\ast}XB + S\right) - XB-S \\ z\left(B^{\ast}X\left(A-I_{n}\right)+S^{\ast}\right) - S^{\ast} & z\left(B^{\ast}XB + R\right) - R \end{array}\right] \\ = & \left[\begin{array}{@{}cc@{}} zI_{n} - A & -B \\ z\left(\left(I_{n}-A^{\ast}\right)X+K^{\ast}K\right) - \left(I_{n}-A^{\ast}\right)XA -K^{\ast}K & zK^{\ast}L - \left(I_{n}-A^{\ast}\right)XB -K^{\ast}L \\ z\left(-B^{\ast}X+L^{\ast}K\right) - B^{\ast}XA -L^{\ast}K & zL^{\ast}L + B^{\ast}XB - L^{\ast}L \end{array}\right]\\ =&\ Z\left(z\tilde E - \tilde A\right). \end{align*} Now assume that we are in the situation of (b). Then, by (ii), $$ \operatorname{im} \hat{Y}$$ is maximally $$\mathcal{E}$$-neutral for $$ \hat{Y} := \left[\begin{array}{@{}c@{}} \hat{Y}_{\mu}\\ \hat{Y}_{x}\\ \hat{Y}_{u} \end{array}\right] =V Y \left[\begin{array}{@{}cc@{}} C & C_{c} \end{array}\right] $$ and V and $$\mathcal{E}$$ as in (30) and (31). By property (i) we obtain $$ \operatorname{rk}\hat{Y}_{x} = \operatorname{rk} \left[\begin{array}{@{}cc@{}} I_{n} & 0 \end{array}\right] \left[\begin{array}{@{}c@{}} \hat{Y}_{x}\\ \hat{Y}_{u} \end{array}\right] = \operatorname{rk} \left[\begin{array}{@{}cc@{}} I_{n} -A & -B \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right] \left[\begin{array}{@{}cc@{}} C & C_{c} \end{array}\right] =n. $$ Thus, there exists an invertible $$T_{1}\in \mathbb{K}^{(n+m)\times (n+m)}$$ such that $$ \hat{Y} T_{1}= \left[\begin{array}{@{}cc@{}} \hat{Y}_{\mu_{1}} & \hat{Y}_{\mu_{2}}\\ I_{n} & 0 \\ \hat{Y}_{u_{1}} & \hat{Y}_{u_{2}} \end{array}\right]. $$ Thus, $$\hat{Y}T_{1}$$ is still maximally $$\mathcal{E}$$-neutral and we obtain $$ 0=\big(\hat{Y}T_{1}\big)^{\ast}\mathcal{E}\hat{Y}T_{1}= \left[\begin{array}{@{}ccc@{}} I_{n} & - \hat{Y}_{\mu_{1}}^{\ast} & 0 \\ 0 & - \hat{Y}_{\mu_{2}}^{\ast} & 0 \end{array}\right] \hat{Y} T_{1} = \left[\begin{array}{@{}cc@{}} \hat{Y}_{\mu_{1}} - \hat{Y}_{\mu_{1}}^{\ast} & \hat{Y}_{\mu_{2}} \\ -\hat Y_{\mu_{2}}^{\ast} & 0 \end{array}\right]; $$ in particular $$X:=-\hat{Y}_{\mu _{1}}$$ is Hermitian. Hence, maximal $$\mathcal{E}$$-neutrality implies full rank of $$\hat{Y}_{u_{2}}$$. Applying another column transformation to $$\hat{Y}$$ via an invertible $$T_{2}\in \mathbb{K}^{(n+m)\times (n+m)}$$ yields $$ \hat{Y} T_{1} T_{2}= \left[\begin{array}{@{}cc@{}} -X & 0 \\ I_{n} & 0 \\ 0 & I_{m} \end{array}\right]. $$ Doing the backtransformation for Y we obtain $$ Y= V^{-1}\hat{Y} T_{1} T_{2} \left[\begin{array}{@{}cc@{}} C & C_{c} \end{array}\right]^{-1} \hat{T}, $$ where $$ \hat{T}:= \left[\begin{array}{@{}cc@{}} C & C_{c} \end{array}\right] (T_{1} T_{2})^{-1} \left[\begin{array}{@{}cc@{}} C & C_{c} \end{array}\right]^{-1}. $$ This implies $$ Y\hat{T}^{-1} = \left[\begin{array}{@{}cc@{}} X(A-I_{n}) & XB \\ I_{n} & 0 \\ 0 & I_{m} \end{array}\right]. $$ We partition $$z\hat{E} - \hat{A} := \left (z\tilde E - \tilde A\right )\hat{T}^{-1}$$ into $$ z\hat{E} - \hat{A} = \left[\begin{array}{@{}cc@{}} z \hat{E}_{1} - \hat{A}_{1} & z \hat{E}_{2} - \hat{A}_{2} \end{array}\right], $$ where $$z \hat{E}_{1} - \hat{A}_{1}\in \mathbb{K}[z]^{(n+q) \times n}$$ and $$z \hat{E}_{2} - \hat{A}_{2}\in \mathbb{K}^{(n+q)\times m}$$. Then, property (iii) implies \begin{equation*} \left[\begin{array}{@{}cc@{}} zI_{n} -A & -B \\ z\left(A^{\ast}X\left(A-I_{n}\right) + Q\right) - X\left(A-I_{n}\right) -Q & z\left(A^{\ast}XB +S\right) - XB - S \\ z\left(B^{\ast}X\left(A-I_{n}\right) + S^{\ast}\right) - S^{\ast} & z\left(B^{\ast}XB +R\right) - R \end{array}\right] = \left[\begin{array}{@{}c@{}} Z_{\mu}\\ Z_{x}\\ Z_{u} \end{array}\right] \left[\begin{array}{@{}cc@{}} z \hat{E}_{1} - \hat{A}_{1} & z \hat{E}_{2} - \hat{A}_{2} \end{array}\right], \end{equation*} yielding $$I_{n} = Z_{\mu } \hat{E}_{1}$$ and thus rk Zμ = n. Therefore, there exists invertible $$T_{3}\in \mathbb{K}^{(n+m)\times (n+m)}$$ such that ZμT3 = [ In 0 ]. Then, for $$ ZT_{3}=: \left[\begin{array}{@{}cc@{}} I_{n} &0\\ Z_{x_{1}} &Z_{x_{2}}\\ Z_{u_{1}} &Z_{u_{2}} \end{array}\right],\qquad T_{3}^{-1} \left( z\hat{E} - \hat{A} \right)=: \left[\begin{array}{@{}cc@{}} z \hat{E}_{11} - \hat{A}_{11} & z \hat{E}_{12} - \hat{A}_{12}\\ z \hat{E}_{21} - \hat{A}_{21} & z \hat{E}_{22} - \hat{A}_{22} \end{array}\right] $$ partitioned accordingly, we obtain \begin{multline} \begin{split}& \left[\begin{array}{@{}cc@{}} zI_{n} -A & -B \\ z\left(A^{\ast}X\left(A-I_{n}\right) + Q\right) - X\left(A-I_{n}\right) -Q & z\left(A^{\ast}XB +S\right) - XB - S \\ z\left(B^{\ast}X\left(A-I_{n}\right) + S^{\ast}\right) - S^{\ast} & z\left(B^{\ast}XB +R\right) - R \end{array}\right]\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=\left[\begin{array}{@{}cc@{}} I_{n} &0\\ Z_{x_{1}} &Z_{x_{2}}\\ Z_{u_{1}} &Z_{u_{2}} \end{array}\right] \left[\begin{array}{@{}cc@{}} z \hat{E}_{11} - \hat{A}_{11} & z \hat{E}_{12} - \hat{A}_{12}\\ z \hat{E}_{21} - \hat{A}_{21} & z \hat{E}_{22} - \hat{A}_{22} \end{array}\right].\end{split} \end{multline} (32) Thus, the first equation gives $$\hat{E}_{11}=I_{n}$$, $$\hat{A}_{11}=A$$, $$\hat{E}_{12}=0$$ and $$\hat{A}_{12}=B$$. For z = 1 we obtain from (32) that $$ \left[\begin{array}{@{}cc@{}} I_{n} - A & -B \\ \left(A^{\ast}-I_{n}\right)X\left(A-I_{n}\right) & \left(A^{\ast}-I_{n}\right)XB \\ B^{\ast}X\left(A-I_{n}\right) & B^{\ast}XB \end{array}\right]\\ = \left[\begin{array}{@{}cc@{}} I_{n} &0\\ Z_{x_{1}} &Z_{x_{2}}\\ Z_{u_{1}} &Z_{u_{2}} \end{array}\right] \left[\begin{array}{@{}cc@{}} I_{n} - A & -B\\ \hat{E}_{21} -\hat{A}_{21} & \hat{E}_{22} - \hat{A}_{22} \end{array}\right]. $$ Multiplying from the right with C results in $$ \left[\begin{array}{@{}cc@{}} \left(I_{n}-A^{\ast}\right)X & 0 \\ 0 &-B^{\ast}X \end{array}\right] \left[\begin{array}{@{}cc@{}} I_{n}-A & -B\\ I_{n}-A & -B \end{array}\right] C = \left[\begin{array}{@{}c@{}} Z_{x_{1}}\\ Z_{u_{1}} \end{array}\right] + \left[\begin{array}{@{}c@{}} Z_{x_{2}}\\ Z_{u_{2}} \end{array}\right] \left[\begin{array}{@{}cc@{}} \hat{E}_{21} -\hat{A}_{21} & \hat{E}_{22} - \hat{A}_{22} \end{array}\right] C $$ and thus $$ \left[\begin{array}{@{}c@{}} Z_{x_{1}}\\ Z_{u_{1}} \end{array}\right]= \left[\begin{array}{@{}c@{}} \left(I_{n}-A^{\ast}\right)X\\ -B^{\ast}X \end{array}\right] - \left[\begin{array}{@{}c@{}} Z_{x_{2}}\\ Z_{u_{2}} \end{array}\right] \left[\begin{array}{@{}cc@{}} \hat{E}_{21} -\hat{A}_{21} & \hat{E}_{22} - \hat{A}_{22} \end{array}\right] C. $$ Inserting this relation into (32) for $$z=\infty $$ gives \begin{align*}& \left[\begin{array}{@{}cc@{}} A^{\ast}X\left(A-I_{n}\right) + Q & A^{\ast}XB + S \\ B^{\ast}X\left(A-I_{n}\right) + S^{\ast} & B^{\ast}XB + R \end{array}\right]\\&\qquad\qquad\qquad = \left[\begin{array}{@{}c@{}} Z_{x_{2}}\\ Z_{u_{2}} \end{array}\right] \left[\begin{array}{@{}cc@{}} \hat{E}_{21} & \hat{E}_{22} \end{array}\right] +\left( \left[\begin{array}{@{}c@{}} \left(I_{n}-A^{\ast}\right)X\\ -B^{\ast}X \end{array}\right] - \left[\begin{array}{@{}c@{}} Z_{x_{2}}\\ Z_{u_{2}} \end{array}\right] \left[\begin{array}{@{}cc@{}} \hat{E}_{21} -\hat{A}_{21} & \hat{E}_{22} - \hat{A}_{22} \end{array}\right] C\right) \left[\begin{array}{@{}cc@{}} I_{n} & 0 \end{array}\right], \end{align*} which leads to \begin{equation*} \mathcal M (X) = \left[\begin{array}{@{}cc@{}} A^{\ast}XA-X + Q & A^{\ast}XB + S \\ B^{\ast}XA + S^{\ast} & B^{\ast}XB + R \end{array}\right]= \left[\begin{array}{@{}c@{}} Z_{x_{2}}\\ Z_{u_{2}} \end{array}\right]\left( \left[\begin{array}{@{}cc@{}} \hat{E}_{21} & \hat{E}_{22} \end{array}\right] - \left[\begin{array}{@{}cc@{}} \hat{E}_{21} -\hat{A}_{21} & \hat{E}_{22} - \hat{A}_{22} \end{array}\right] \left[\begin{array}{@{}cc@{}} C & 0 \end{array}\right] \right). \end{equation*} Thus, we have \begin{align} \operatorname{rk} \mathcal M (X) \le q. \end{align} (33) Further, by Lemma 3.2, for ω ∈ [0, 2π) we can rewrite $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )$$ as $$ \Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right)=\left[\begin{array}{@{}c@{}} \left(\mathrm{e}^{\mathrm{i}\omega} I_{n}-A\right)^{-1}B \\ I_{m} \end{array}\right]^{\ast} \mathcal{M}(X) \left[\begin{array}{@{}c@{}} \left(\mathrm{e}^{\mathrm{i}\omega} I_{n}-A\right)^{-1}B \\ I_{m} \end{array}\right] $$ and thus in (33) we even have equality. Therefore, we can apply Lemma 6.2 and hence, we have shown that (a) holds. In the case of a BVD pencil we can prove a similar statement. Theorem 6.5 Let $$(I_{n},A,B,Q,S,R) \in \Sigma^w_{m,n}(\mathbb{K})$$ be given and consider the associated BVD pencil $$z\mathcal{E}-\mathcal{A}$$ as in (12). Further, let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. Then, the following are equivalent: (a) There exists a solution $$(X,\,K,\,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ of (27). (b) It holds that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π) such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} I_{n}-A\right )\neq 0$$. Furthermore, there exist matrices $$Y_{\mu },\,Y_{x}\in \mathbb{K}^{n\times (n+m)},\,Y_{u}\in \mathbb{K}^{m\times (n+m)}$$ and $$Z_{\mu },\,Z_{x}\in \mathbb{K}^{n\times (n+q)},\,Z_{u}\in \mathbb{K}^{m\times (n+q)}$$ such that for $$ Y=\left[\begin{array}{@{}c@{}} Y_{\mu}\\ Y_{x}\\ Y_{u} \end{array}\right],\qquad Z=\left[\begin{array}{@{}c@{}} Z_{\mu}\\ Z_{x}\\ Z_{u} \end{array}\right] $$ the following holds: (i) The matrix $$ Y_{x}= \left[\begin{array}{@{}cc@{}} I_{n} & 0 \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right] $$ has full row rank n. (ii) The space $$\mathcal Y= \operatorname{im} Y$$ is maximally $$\mathcal{E}^{e}$$-neutral, where $$ \mathcal{E}^{e}:= \left[\begin{array}{@{}ccc@{}} 0 & -I_{n} & 0\\ I_{n}& 0 & 0\\ 0 & 0 & 0 \end{array}\right]. $$ (iii) There exist $$\tilde E,\,\tilde A \in \mathbb{K}^{(n+q)\times (n+m)}$$ such that $$(z\mathcal{E}-\mathcal{A})Y=Z(z\tilde E - \tilde A)$$. Proof. See the thesis by Bankmann (2016, Theorem 5.5). 6.2. Implicit difference equations In this section we generalize the results from the previous section to IDEs. As for the KYP inequality we need relations between the Lur’e equation (25) corresponding to the original system and the associated equation corresponding to the feedback equivalent system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17). These findings are related to the results in the article by Reis et al. (2015) in the continuous-time case. Lemma 6.6 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. Then, $$(X\,,K\,,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ is a solution of (25) if and only if \begin{align} \left(X_{F},\,K_{F},\,L_{F}\right) := \left(W^{-*}XW^{-1},\,KT+LFT,\,L\right) \end{align} (34) is a solution of (25) associated to the feedback system $$ (E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17), i.e. $$ \mathcal{M}_{F}\left(X_{F}\right)= \left[\begin{array}{@{}cc@{}} A_{F}^{\ast}X_{F}A_{F}-E_{F}^{\ast}X_{F}E_{F} + Q_{F} & A_{F}^{\ast}X_{F}B_{F} + S_{F} \\ B^{\ast}_{F}X_{F}A_{F} + S_{F}^{\ast} & B_{F}^{\ast}X_{F}B_{F} + R_{F} \end{array}\right] =\,_{\mathcal{V}_{(E_F,A_F,B_F)}} \left[\begin{array}{@{}c@{}} K_{F}^{\ast}\\ L_{F}^{\ast} \end{array}\right] \left[\begin{array}{@{}cc@{}} K_{F} & L_{F} \end{array}\right]. $$ Proof. First note that for $$ \mathcal{T}_{F} = \left[\begin{array}{@{}cc@{}} T & 0 \\ FT & I_{m} \end{array}\right] $$ we have $$ \left[\begin{array}{@{}cc@{}} K & L \end{array}\right] \mathcal{T}_{F} = \left[\begin{array}{@{}cc@{}} KT + LFT & L \end{array}\right] . $$ In addition, by Proposition 3.4(a) we obtain that $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)=\textrm{rk}_{\mathbb{K}(z)} \Phi _{F}(z)$$. Thus, Lemma 3.3 immediately yields the assertion. Moreover, we now characterize the connection between the Lur’e equation (25) corresponding to the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ in feedback equivalence form as in (17) and the Lur’e equation (27) corresponding to the associated EDE part as in (18). Lemma 6.7 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and consider the system $$(E_{F},A_{F},B_{F},Q_{F},S_{F}, R_{F})\in\Sigma^{w}_{m,n}(\mathbb{K})$$ as in (17) in feedback equivalence form (3). Further, consider (XF, KF, LF) as in (34) partitioned according to the block structure of the feedback equivalence form. Then, with $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$ we have that $$(X\,,K\,,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ is a solution of (25) if and only if $$(X_{11},\,K_{1},\,L-K_{2}B_{2})\in \mathbb{K}^{n_{1}\times n_{1}}\times \mathbb{K}^{q\times n_{1}}\times \mathbb{K}^{q\times m}$$ is a solution of (27) for the EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18). Proof. See the study by Bankmann (2016, Lemma 5.7). For the rest of this chapter we assume that $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ is I-controllable, i.e. there exists a feedback $$F\in \mathbb{K}^{m\times n}$$ such that the system $$ (E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) is in feedback equivalence form such that n3 = 0. This is justified by the fact that the subsystem described by $$(E_{33},I_{n_{3}},0) \in \Sigma_{m,n_{3}}(\mathbb{K})$$ obtained from the feedback equivalence form (3) has only the zero solution and thus does not contribute to the dynamics of the system. Indeed, in the proofs of Lemma 3.5 and Lemma 6.7 the parts of $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ corresponding to the last n3 variables do not contribute to the analysis. The following proposition makes this precise, using the same projection ansatz as in the study of Reis et al. (2015, Theorem 5.9). Proposition 6.8 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and consider the system $$(E_F,A_F,B_F,Q_F, S_F,R_F)\in\Sigma^{w}_{m,n}(\mathbb{K}) $$ as in (17) in feedback equivalence form (3). Further, let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. Define the projector $$ \varPi:= W^{-1} \left[\begin{array}{@{}ccc@{}} I_{n_{1}} & 0 & 0\\ 0 &0 &0\\ 0 & 0 &0 \end{array}\right] W \in\mathbb{K}^{n \times n}. $$ Then, we have \begin{align} \operatorname{im} \varPi = E\mathcal{W}_{(E,A,B)} \end{align} (35) and the following statements hold: (a) The system $$(\varPi E, A, B)\in \Sigma_{m,n}(\mathbb{K})$$ is I-controllable and $$ \mathfrak{B}_{(E,A,B)}=\mathfrak{B}_{(\varPi E,A,B)}.$$ In particular, the system space of $$(\varPi E, A, B)$$ is $$\mathcal{V}_{(E,A,B)} $$. (b) There exists a solution $$P\in \mathbb{K}^{n \times n}$$ of the KYP inequality (16), i.e. $$\mathcal M (P)\succeq _{\mathcal{V}_{(E,A,B)} }0$$, if and only if $$\mathcal M_{\varPi } (P)\succeq _{\mathcal{V}_{(E,A,B)} }0$$, where $$\mathcal{M}_{\varPi }(P)$$ is the matrix in (16) corresponding to the system $$(\varPi E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$. (c) There exists a solution $$(X,\,K,\,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ of the Lur’e equation (25) if and only if (X, K, L) also fulfils the Lur’e equation (25) corresponding to the system $$(\varPi E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$. Proof. Part (a) and (35) follow with the algebraic manipulations mentioned in the proof of Reis et al. (2015, Theorem 5.9). Now set $$ \varPi_{F}:= W\varPi W^{-1}= \left[\begin{array}{@{}ccc@{}} I_{n_{1}} & 0 & 0\\ 0 &0 &0\\ 0 & 0 &0 \end{array}\right]. $$ For parts (b) and (c) note that the system $$(\varPi _{F}E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is in feedback equivalence form (3) where compared to $$(E_{F}, A_{F}, B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ the matrices E23 and E33 are set to zero. Looking carefully at the proofs of Lemma 3.5 and Lemma 6.7 we see that these matrices have no effect in the respective results and thus the assertion follows. As a next step, we perform transformations of the palindromic or BVD pencils corresponding to the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ in feedback equivalence form as in (17) such that we obtain the respective palindromic or BVD pencils corresponding to the EDE system $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18) in the first diagonal block of the transformed pencil. Lemma 6.9 Let $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) be given in feedback equivalence form (3) such that n3 = 0. Further, let the corresponding palindromic pencil $$z\mathcal{A}_{F}^{\ast }-\mathcal{A}_{F}$$ as in (14) be given. Denote by $$z\mathcal{A}_{s}^{\ast }-\mathcal{A}_{s}$$ the pencil corresponding to the EDE system $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18), i.e. \begin{align} z\mathcal{A}_{s}^{\ast}-\mathcal{A}_{s}= \left[\begin{array}{@{}ccc@{}} 0 & zI_{n_{1}}-A_{s} & -B_{s} \\ zA_{s}^{\ast}-I_{n_{1}} & (z-1)Q_{s} & (z-1)S_{s} \\ zB_{s}^{\ast} & (z-1)S_{s}^{\ast} & (z-1)R_{s} \end{array}\right]\in\mathbb{K}[z]^{(2n_{1}+m)\times (2n_{1}+m)}. \end{align} (36) Then, there exists an invertible $$\hat{U}\in \mathbb{K}^{(2n+m)\times (2n+m)}$$ such that $$ \hat{U}^{\ast} \left(z\mathcal{A}_{F}^{\ast}-\mathcal{A}_{F}\right) \hat U = \left[ \begin{array}{@{}c|cc@{} } z\mathcal{A}_{s}^{\ast}-\mathcal{A}_{s} & zD & 0 \\\hline -D^{\ast} & 0 & zI_{n_{2}} \\ 0 & -I_{n_{2}} & 0 \end{array}\right], $$ where $$ D= \left[\begin{array}{@{}c@{}} 0 \\ Q_{12} \\ S_{2}^{\ast}-B_{2}^{\ast}Q_{22} \end{array}\right] $$ and \begin{align} \hat U&= \left[\begin{array}{@{}ccccc@{}} I_{n_{1}} & 0 & 0 & 0 & 0 \\ 0 & -Q_{12}^{\ast} & -S_{2} +Q_{22}B_{2} & -Q_{22} & I_{n_{2}} \\ 0 & I_{n_{1}} & 0 & 0 & 0 \\ 0 & 0 & -B_{2} & I_{n_{2}} & 0 \\ 0 & 0 & I_{m} & 0 & 0\nonumber \end{array}\right]\\ &= \underbrace{ \left[\begin{array}{@{}ccccc@{}} I_{n_{1}} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & I_{n_{2}} \\ 0 & I_{n_{1}} & 0 & 0 & 0 \\ 0 & 0 & 0 &I_{n_{2}}& 0 \\ 0 & 0 & I_{m} & 0 & 0 \end{array}\right]}_{=:\tilde P} \underbrace{ \left[\begin{array}{@{}ccccc@{}} I_{n_{1}} & 0 & 0 & 0 & 0 \\ 0 & I_{n_{1}} & 0 & 0 & 0 \\ 0 & 0 & I_{m} & 0 & 0 \\ 0 & 0 & -B_{2} & I_{n_{2}} & 0 \\ 0 & -Q_{12}^{\ast} & -S_{2}+Q_{22}B_{2}& -Q_{22} & I_{n_{2}} \end{array}\right].}_{=:\tilde U} \end{align} (37) Proof. See the thesis by Bankmann (2016, Lemma 5.9). Corollary 6.10 Let $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) be given in feedback equivalence form (3) such that n3 = 0. Further, let the corresponding BVD pencil $$z\mathcal{E}_{F}-\mathcal{A}_{F}$$ as in (12) be given. Denote by $$z\mathcal{E}_{s}-\mathcal{A}_{s}$$ the BVD pencil corresponding to the EDE system $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18), i.e. $$ z\mathcal{E}_{s}-\mathcal{A}_{s}= \left[\begin{array}{@{}ccc@{}} 0 & zI_{n_{1}}-A_{s} & -B_{s} \\ zA_{s}^{\ast}-I_{n_{1}} & -Q_{s} & -S_{s} \\ zB_{s}^{\ast} & -S_{s}^{\ast} & -R_{s} \end{array}\right]\in\mathbb{K}[z]^{(2n_{1}+m)\times (2n_{1}+m)}. $$ Then, there exist invertible $$\hat{U}$$ and $$\check U$$ such that $$ \hat{U}^{\ast} (z\mathcal{E}-\mathcal{A}) \check U = \left[ \begin{array}{@{}c|cc@{} } z\mathcal{E}_{s}-\mathcal{A}_{s} & 0 & 0\\ \hline -D^{\ast} & 0 & zI_{n_{2}} \\ 0 & -I_{n_{2}} & 0 \end{array}\right]. $$ Now we are able to prove a generalization of Theorem 6.4. This result is related to the result in the article by Reis et al. (2015, Theorem 6.2) in the continuous-time case. Theorem 6.11 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be I-controllable. Further, let the corresponding palindromic pencil $$z\mathcal{A}^{\ast }-\mathcal{A}$$ (14) be given. In addition, let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$ and assume that $$ \operatorname{rk}\,\left [ \,E-A \quad B \,\right ] =n$$. Then, the following are equivalent: (a) There exists a solution $$(X,\,K,\,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ of the Lur’e equation (25). (b) It holds that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π) such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. Furthermore, there exist matrices $$Y_{\mu },\,Y_{x}\in \mathbb{K}^{n\times (n+m)},\,Y_{u}\in \mathbb{K}^{m\times (n+m)}$$ and $$Z_{\mu },\,Z_{x}\in \mathbb{K}^{n\times (n+q)},\,Z_{u}\in \mathbb{K}^{m\times (n+q)}$$ such that for $$ Y=\left[\begin{array}{@{}c@{}} Y_{\mu}\\ Y_{x}\\ Y_{u} \end{array}\right],\qquad Z=\left[\begin{array}{@{}c@{}} Z_{\mu}\\ Z_{x}\\ Z_{u} \end{array}\right] $$ the following holds: (i) The matrix $$ \left[\begin{array}{@{}cc@{}} E-A & -B \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right] $$ has rank n1. (ii) The space $$\mathcal Y= \operatorname{im} Y$$ is of dimension n + m and $$\left (\mathcal{A}^{\ast }-\mathcal{A}\right )$$-neutral. (iii) It holds that $$ \mathcal{V}_{(E,A,B)}= \operatorname{im} \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right]. $$ (iv) There exist $$\tilde E,\,\tilde A \in \mathbb{K}^{(n+q)\times (n+m)}$$ such that $$\left (z\mathcal{A}^{\ast }-\mathcal{A}\right )Y=Z\left (z\tilde E - \tilde A\right )$$. Proof. First we show that the statement is invariant under feedback transformations. Therefore, assume we have given the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ in feedback equivalence form as in (17) such that n3 = 0 with corresponding transformation matrices W and $$\mathcal T_{F}$$ and corresponding palindromic pencil $$z\mathcal{A}_{F}^{\ast } -\mathcal{A}_{F}$$ as in (14). Then, by Lemma 6.6, part (a) is equivalent to the existence of a solution (XF, KF, LF) as in (34) of the Lur’e equation (25) corresponding to $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$. To show the equivalence of statement (b) to according statements for the system in feedback equivalence form let $$ U_{F}:= \left[\begin{array}{@{}ccc@{}} W^{\ast} & 0 & 0 \\ 0 & T & 0 \\ 0 & FT & I_{m} \end{array}\right]\in\mathbb{K}^{(2n+m)\times (2n+m)} $$ and set \begin{align} Y_{F}:=\left[\begin{array}{@{}c@{}} Y_{\mu,F}\\ Y_{x,F}\\ Y_{u,F} \end{array}\right] := U_{F}^{-1} Y ,\quad Z_{F}:=\left[\begin{array}{@{}c@{}} Z_{\mu,F}\\ Z_{x,F}\\ Z_{u,F} \end{array}\right] = U_{F}^{*} Z . \end{align} (38) Then, $$ \mathcal{A}_{F} = U_{F}^{*}\mathcal{A} U_{F}$$ and statement (i) is equivalent to \begin{equation*} \operatorname{rk} \left[\begin{array}{@{}cc@{}} E_{F}-A_{F} & -B_{F} \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x,F}\\ Y_{u,F} \end{array}\right] = \operatorname{rk} \left[\begin{array}{@{}cc@{}} E-A & -B \end{array}\right] \left[\begin{array}{@{}cc@{}} T & 0 \\ FT & I_{m} \end{array}\right] \left[\begin{array}{@{}cc@{}} T & 0 \\ FT & I_{m} \end{array}\right]^{-1} \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right]=n_{1}. \end{equation*} Furthermore, we have that rkYF = rkY = n + m and imY is $$\left (\mathcal{A}^{\ast }-\mathcal{A}\right )$$-neutral if and only if imYF is $$\left (\mathcal{A}_{F}^{\ast }-\mathcal{A}_{F}\right )$$-neutral. In addition, by Proposition 2.11(a) we obtain that (iii) is equivalent to $$ \mathcal{V}_{(E_F,A_F,B_F)}= \operatorname{im} \left[\begin{array}{@{}c@{}} Y_{x,F}\\ Y_{u,F} \end{array}\right]. $$ Finally, statement (iv) is equivalent to $$\left (z\mathcal{A}_{F}^{\ast }-\mathcal{A}_{F}\right )Y_{F}=Z_{F}\left (z\tilde E - \tilde A\right )$$ by the definition of $$\mathcal{A}_{F},\,Y_{F}\,$$ and ZF. Hence, we have shown that it is sufficient to prove the equivalence between (a) and (b) for the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form. Now we show that statement (b) follows from statement (a). From Lemma 6.7 we infer that $$\left (X_{11},\,K_{1},\,L-K_{2}B_{2}\right )$$ is a solution of the Lur’e equation (27) for the EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18). By denoting the corresponding palindromic pencil arising in the optimal control problem by $$z\mathcal{A}_{s}^{\ast } - \mathcal{A}_{s}$$ as in (36), Theorem 6.4 implies the existence of $$ Y_{s}= \left[\begin{array}{@{}cc@{}} X_{11}\left(A_{11}-I_{n_{1}}\right) & X_{11}B_{1} \\ I_{n_{1}} & 0 \\ 0 & I_{m} \end{array}\right], \qquad Z_{s} = \left[\begin{array}{@{}cc@{}} I_{n_{1}} & 0 \\ \left(I_{n_{1}}-A_{11}^{\ast}\right)X_{11} & K_{1}^{\ast} \\ -B_{1}^{\ast}X_{11} & \left(L-K_{2}B_{2}\right)^{\ast} \end{array}\right] $$ as in (28) and $$ z\hat{E}_{s} - \hat{A}_{s}= \left[\begin{array}{@{}cc@{}} zI_{n_{1}}-A_{11} & -B_{1} \\ (z-1)K_{1} &(z-1)\left(L-K_{2}B_{2}\right) \end{array}\right] $$ as in (29) such that $$\left (z\mathcal{A}_{s}^{\ast }-\mathcal{A}_{s}\right )Y_{s}=Z_{s}\left (z\hat{E}_{s} - \hat{A}_{s}\right )$$. Note that as in Theorem 6.4, im Ys is maximally $$\left (\mathcal{A}_{s}^{\ast }-\mathcal{A}_{s}\right )$$-neutral. From Lemma 6.9 we obtain an invertible transformation matrix $$\hat{U}\in \mathbb{K}^{(2n+m)\times (2n+m)}$$ as in (37) such that \begin{align} z\hat{\mathcal{A}}^{\ast}-\hat{\mathcal{A}} := \hat{U}^{\ast} \left(z\mathcal{A}_{F}^{\ast}-\mathcal{A}_{F}\right) \hat{U} = \left[ \begin{array}{@{}c|cc@{} } z\mathcal{A}_{s}^{\ast}-\mathcal{A}_{s} & zD & 0\\ \hline -D^{\ast} & 0 & zI_{n_{2}} \\ 0 & -I_{n_{2}} & 0 \end{array}\right] \end{align} (39) with $$ \ D= \left[\begin{array}{@{}c@{}} 0 \\ Q_{12} \\ S_{2}^{\ast}-B_{2}^{\ast}Q_{22} \end{array}\right]\in\mathbb{K}^{(2n_{1}+m)\times n_{2}}. $$ By inspecting the proof of Theorem 6.4 we find that \begin{align} \left(z\hat{\mathcal{A}}^{\ast}-\hat{\mathcal{A}}\right) \hat{Y} = \hat{Z} \left(z\hat{E} - \hat{A}\right), \end{align} (40) where $$ \hat{Y} = \left[ \begin{array}{@{}c|c@{} } Y_{s} & 0 \\\hline 0 & 0 \\ 0 & I_{n_{2}} \end{array}\right], \qquad \hat{Z} =\left[ \begin{array}{@{}c|c@{} } Z_{s} & 0\\\hline 0 &I_{n_{2}}\\ 0 & 0 \end{array}\right],\qquad z\hat{E} - \hat{A}=\left[ \begin{array}{@{}c|c@{} } z\hat{E}_{s}-\hat{A}_{s} & 0\\ \hline -D^{\ast}Y_{s} & zI_{n_{2}} \end{array}\right]. $$ Thus, we have $$ \hat{Y}^{\ast}\left(\hat{\mathcal{A}}^{\ast}-\hat{\mathcal{A}}\right)\hat{Y} = \left[ \begin{array}{@{}c|cc@{} } Y_{s}^{\ast} & 0 & 0 \\\hline 0 & 0 & I_{n_{2}} \end{array}\right] \left[ \begin{array}{@{}c|cc@{} } \mathcal{A}_{s}^{\ast}-\mathcal{A}_{s} & D & 0\\ \hline -D^{\ast} & 0 & I_{n_{2}} \\ 0 & -I_{n_{2}} & 0 \end{array}\right] \left[ \begin{array}{@{}c|c@{} } Y_{s} & 0\\ \hline 0 & 0 \\ 0 & I_{n_{2}} \end{array}\right]=0, $$ and we obtain that $$ \operatorname{im} \hat{Y}$$ is n + m dimensional and $$\left (\hat{\mathcal{A}}^{\ast }-\hat{\mathcal{A}}\right )$$-neutral. Set $$ \hat{V} = \left[\begin{array}{@{}ccc@{}} I_{n_{1}} & 0 & 0 \\ 0 & 0 & I_{m} \\ Q_{12}^{\ast} & -I_{n_{2}} & -B_{2} + S_{2}-Q_{22}B_{2} \end{array}\right]. $$ Transforming the quantities in (40) to feedback equivalence form (17) we obtain $$ \left(z\mathcal{A}_{F}^{\ast}-\mathcal{A}_{F}\right) Y_{F}\hat{V} = Z_{F} \left(z\tilde E - \tilde A\right), $$ where \begin{align} Y_{F} \hat{V}= \left[\begin{array}{@{}c@{}} Y_{\mu,F}\\ Y_{x,F}\\ Y_{u,F} \end{array}\right]\hat{V} := \left[\begin{array}{@{}c@{}} Y_{\mu_{1},F}\\ Y_{\mu_{2},F}\\ Y_{x_{1},F}\\ Y_{x_{2},F}\\ Y_{u,F} \end{array}\right]:= \hat{U}\hat{Y}\hat{V}=\left[ \begin{array}{@{}ccc@{} } X_{11}(A_{11}-I_{n_{1}}) &0 & X_{11}B_{1} \\ 0 & -I_{n_{2}} & -B_{2}\\ \hline I_{n_{1}} & 0 & 0 \\ 0 & 0 & -B_{2} \\ 0 & 0 & I_{m} \end{array}\right], \end{align} (41)$$Z_{F}:= \hat{U}^{-*}\hat{Z}$$, and $$ \left(z\tilde E - \tilde A\right):=\left(z\hat{E} - \hat{A}\right)\hat{V}=\left[ \begin{array}{@{}ccc@{} } zI_{n_{1}}-A_{11} & 0 & -B_{1} \\ (z-1)K_{1} & 0 & (z-1)\left(L-K_{2}B_{2}\right) \\ (z-1)Q_{12}^{\ast} & -zI_{n_{2}} & -zB_{2} +(z-1) \left(S_{2} - Q_{22}B_{2} \right) \end{array}\right]. $$ Then, we obtain property (i) by \begin{align*} \operatorname{rk} \left[\begin{array}{@{}cc@{}} E_{F}-A_{F} & -B_{F} \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x,F}\\ Y_{u,F} \end{array}\right] &= \operatorname{rk} \left[\begin{array}{@{}ccc@{}} I_{n_{1}}-A_{11} & 0 & -B_{1}\\ 0 &-I_{n_{2}} & -B_{2} \end{array}\right] \left[\begin{array}{@{}ccc@{}} I_{n_{1}} & 0 & 0 \\ 0 & -B_{2} & 0 \\ 0 & I_{m} & 0 \end{array}\right]\\ &= \operatorname{rk} \left[\begin{array}{@{}ccc@{}} I_{n_{1}}-A_{11} & -B_{1} & 0\\ 0 & 0 & 0 \end{array}\right] =n_{1}. \end{align*} Property (ii) follows from the fact that $$ \operatorname{im} \hat{Y}$$ is n + m dimensional and $$\left (\hat{\mathcal{A}}^{\ast }-\hat{\mathcal{A}}\right )$$-neutral. Furthermore, by Proposition 2.11(a) we have property (iii). Altogether, this shows statement (b). Now assume that (b) holds for the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form, i.e. properties (i)–(iv) are satisfied. From these properties we construct a deflating subspace for the palindromic pencil $$z\mathcal{A}_{s}^{\ast }-\mathcal{A}_{s}$$ such that we can apply Theorem 6.4. Therefore, with the help of Proposition 2.11(a) and (iii) we accordingly partition YF into $$ Y_{F} = \left[\begin{array}{@{}c@{}} Y_{\mu,F}\\ Y_{x,F}\\ Y_{u,F} \end{array}\right] := \left[\begin{array}{@{}c@{}} Y_{\mu_{1},F}\\ Y_{\mu_{2},F}\\ Y_{x_{1},F}\\ Y_{x_{2},F}\\ Y_{u,F} \end{array}\right]:= \left[\begin{array}{@{}ccc@{}} Y_{\mu_{11},F} & Y_{\mu_{12},F} & Y_{\mu_{13},F} \\ Y_{\mu_{21},F} & Y_{\mu_{22},F} & Y_{\mu_{23},F}\\ I_{n_{1}} & 0 & 0\\ 0 & -B_{2} & 0\\ 0 & I_{m} & 0 \end{array}\right] $$ and denote by $$\hat{\mathcal{A}}$$ and $$\hat{U}$$ the matrices we obtain from Lemma 6.9 such that (39) holds. Then, for $$\hat{Y} := \hat{U}^{-1} Y_{F}$$ we have $$ \hat{Y}= \hat{U}^{-1} Y_{F} = \left[\begin{array}{@{}ccccc@{}} I_{n_{1}} & 0 & 0 & 0 & 0 \\ 0 & 0 & I_{n_{1}} & 0 & 0 \\ 0 & 0 & 0 & 0 & I_{m} \\ 0 & 0 & 0 & I_{n_{2}} & B_{2} \\ 0 & I_{n_{2}} & Q_{12}^{\ast} & Q_{22} & S_{2} \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{\mu_{1},F}\\ Y_{\mu_{2},F}\\ Y_{x_{1},F}\\ Y_{x_{2},F}\\ Y_{u,F} \end{array}\right] = \left[\begin{array}{@{}c@{}} Y_{\mu_{1},F}\\ Y_{x_{1},F}\\ Y_{u,F}\\ 0\\ \hat{Y}_{\mu_{2},F} \end{array}\right] $$ for some $$ \hat{Y}_{\mu_{2},F}:= \left[\begin{array}{@{}ccc@{}} \hat{Y}_{\mu_{21},F} & \hat{Y}_{\mu_{22},F} & \hat{Y}_{\mu_{23},F} \end{array}\right]. $$ Thus, $$ \operatorname{im} \hat{Y}$$ is n + m dimensional by property (ii) and $$\left (\hat{\mathcal{A}}^{\ast }-\hat{\mathcal{A}}\right )$$-neutral. In particular we obtain \begin{align*} 0&=\! \left[\begin{array}{@{}ccccc@{}}Y_{\mu_{13},F}^{\ast}& 0 & 0 & 0 & \hat{Y}_{\mu_{23},F}^{\ast} \end{array}\right]\! \left[ \begin{array}{@{}ccc|cc@{} } 0 & I_{n_{1}}-A_{11} & -B_{1} & 0 & 0 \\ A_{11}^{\ast}-I_{n_{1}} & 0 & 0 & Q_{12} & 0 \\ B_{1}^{\ast} & 0 & 0 & S_{2}^{\ast} -B_{2}^{\ast}Q_{22}& 0\\ \hline 0 & -Q_{12}^{\ast} & Q_{22}B_{2}^{\ast} - S_{2} & 0 & I_{n_{2}}\\ 0 & 0 & 0 &-I_{n_{2}} & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} Y_{\mu_{11},F} & Y_{\mu_{12},F} \\ I_{n_{1}} & 0 \\ 0 & I_{m} \\ 0&0\\ \hat{Y}_{\mu_{21},F} & \hat{Y}_{\mu_{22},F} \end{array}\right]\\ &= Y_{\mu_{13},F}^{\ast} \left[\begin{array}{@{}cc@{}} I_{n_{1}} - A_{s} & - B_{s} \end{array}\right], \end{align*} and hence $$Y_{\mu _{13},F}=0$$. This shows that the matrix $$ \left[\begin{array}{@{}c@{}} Y_{\mu_{1},F}\\ Y_{x_{1},F}\\ Y_{u,F} \end{array}\right] $$ has full column rank and thus its image is also maximally $$\left (\mathcal{A}_{s}^{\ast }-\mathcal{A}_{s}\right )$$-neutral. This, together with the fact that $$ \operatorname{rk} \hat{Y} = n+m$$, allows us to perform a column transformation of $$\hat{Y}$$ via $$T_{1}\in \mathbb{K}^{(n+m)\times (n+m)}$$ such that $$ \left[\begin{array}{@{}c@{}} Y_{\mu_{1},F}\\ Y_{x_{1},F}\\ Y_{u,F}\\ \hat{Y}_{\mu_{2},F}\\ \hat{Y}_{x_{2},F} \end{array}\right] T_{1}= \hat{Y} T_{1}= \left[ \begin{array}{@{}cc|c@{} } X_{11}\left(A_{11}-I_{n_{1}}\right) & X_{11}B_{1} & 0 \\ I_{n_{1}} & 0 & 0 \\ 0 & I_{m} & 0\\ \hline 0 & 0 & 0 \\ 0 & 0 & I_{n_{2}} \end{array}\right] $$ with some Hermitian $$X_{11}\in \mathbb{K}^{n_{1}\times n_{1}}$$, similar as in the proof of Theorem 6.4. Set $$ Y_{s}:= \left[\begin{array}{@{}cc@{}} X_{11}\left(A_{11}-I_{n_{1}}\right) & X_{11}B_{1} \\ I_{n_{1}} & 0 \\ 0 & I_{m} \end{array}\right]. $$ From property (iv) we obtain \begin{align} \left[ \begin{array}{@{}c|cc@{} } z\mathcal{A}_{s}^{\ast}-\mathcal{A}_{s} & zD & 0\\ \hline -D^{\ast} & 0 & zI_{n_{2}} \\ 0 & -I_{n_{2}} & 0 \end{array}\right] \left[ \begin{array}{@{}c|c@{} } Y_{s} & 0\\ \hline 0 & 0\\ 0 & I_{n_{2}} \end{array}\right]= \left[\begin{array}{@{}cc@{}} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{array}\right] \left[\begin{array}{@{}cc@{}} z\hat{E}_{11}-\hat{A}_{11} & z\hat{E}_{12}-\hat{A}_{12}\\ z\hat{E}_{21}-\hat{A}_{21} & z\hat{E}_{22}-\hat{A}_{22} \end{array}\right], \end{align} (42) where $$Z_{11}\in \mathbb{K}^{(2n_{1}+m)\times (n_{1}+q)}$$, $$Z_{12}\in \mathbb{K}^{(2n_{1}+m)\times n_{2}}$$, $$Z_{21}\in \mathbb{K}^{2n_{2}\times (n_{1}+q)}$$, $$Z_{22}\in \mathbb{K}^{2n_{2}\times n_{2}}$$, $$z\hat{E}_{11}-\hat{A}_{11}\in \mathbb{K}[z]^{(n_{1}+q)\times (n_{1}+m)}$$, $$z\hat{E}_{12}-\hat{A}_{12}\in \mathbb{K}[z]^{(n_{1}+q)\times n_{2}}$$, $$z\hat{E}_{21}-\hat{A}_{21}\in \mathbb{K}[z]^{n_{2}\times (n_{1}+m)}$$ and $$z\hat{E}_{22}-\hat{A}_{22}\in \mathbb{K}[z]^{n_{2}\times n_{2}}$$. From the last block column and block row of (42) we obtain \begin{align} \left[\begin{array}{@{}c@{}} zI_{n_{2}}\\ 0 \end{array}\right]= \left[\begin{array}{@{}cc@{}} Z_{21} & Z_{22} \end{array}\right] \left[\begin{array}{@{}c@{}} z\hat{E}_{12}-\hat{A}_{12}\\ z\hat{E}_{22}-\hat{A}_{22} \end{array}\right] \end{align} (43) and thus we have $$ \operatorname{rk} \left[\begin{array}{@{}cc@{}} Z_{21} &Z_{22} \end{array}\right]=n_{2}. $$ Therefore, we can determine a transformation matrix $$T_{2}\in \mathbb{K}^{(n+q)\times (n+q)}$$ such that $$ \operatorname{rk} \left[\begin{array}{@{}cc@{}} Z_{21} &Z_{22} \end{array}\right]T_{2}= \left[\begin{array}{@{}cc@{}} 0 & \tilde Z_{22} \end{array}\right] $$ for some $$\tilde Z_{22}\in \mathbb{K}^{2n_{2}\times n_{2}}$$. Set $$ \left[\begin{array}{@{}cc@{}} \tilde Z_{11} & \tilde Z_{12} \\ 0 & \tilde Z_{22} \end{array}\right]:= \left[\begin{array}{@{}cc@{}} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{array}\right]T_{2} $$ and $$ \left[\begin{array}{@{}cc@{}} z\tilde E_{11} - \tilde A_{11} & z \tilde E_{12}-\tilde A_{12}\\ z\tilde E_{21} - \tilde A_{21} & z\tilde E_{22}-\tilde A_{22} \end{array}\right]:=T_{2}^{-1} \left[\begin{array}{@{}cc@{}} z\hat{E}_{11}-\hat{A}_{11} & z\hat{E}_{12}-\hat{A}_{12}\\ z\hat{E}_{21}-\hat{A}_{21} & z\hat{E}_{22}-\hat{A}_{22} \end{array}\right], $$ accordingly partitioned. Thus, re-evaluating (43) for the transformed matrices we also obtain full normal rank n2 of $$z\tilde E_{22}-\tilde A_{22}$$. Hence, there exists some $$\lambda _{0}\in \mathbb{C}$$ such that $$\lambda _{0} \tilde E_{22}-\tilde A_{22}$$ is invertible. From the last block column and first block row of (42) we infer $$ 0=\tilde Z_{11}\left(\lambda_{0} \tilde E_{12}-\tilde A_{12}\right) + \tilde Z_{12}\left(\lambda_{0} \tilde E_{22}-\tilde A_{22}\right). $$ Thus, $$\tilde Z_{12}$$ can be expressed as $$ \tilde Z_{12}=-\tilde Z_{11}\left(\lambda_{0} \tilde E_{12}-\tilde A_{12}\right)\left(\lambda_{0} \tilde E_{22}-\tilde A_{22}\right)^{-1}. $$ Inserting this relation into the first block row and block column of (42) we have $$ \left(z\mathcal{A}_{s}^{\ast}-\mathcal{A}_{s}\right)Y_{s} = \tilde Z_{11}\left(z\tilde E_{11}-\tilde A_{11} - \left(\lambda_{0} \tilde E_{12}-\tilde A_{12}\right)\left(\lambda_{0} \tilde E_{22}-\tilde A_{22}\right)^{-1}\left(z\tilde E_{21}-\tilde A_{21}\right) \right). $$ Hence, we are finally in the position to apply Theorem 6.4. From this we obtain a solution (Xs, Ks, Ls) of (27) corresponding to the system $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)$$. By Lemma 6.7 we then also find a solution (XF, KF, LF) of (25) corresponding to the system in feedback equivalence form $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$. Remark 6.12 Let an I-controllable system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and let $$z\mathcal{A}^{\ast }-\mathcal{A}$$ be the palindromic pencil as in (14). Further, assume that there exists a solution (X, K, L) of the Lur’e equation (25). (a) The matrix pencil $$z\tilde E - \tilde A\in \mathbb{K}[z]^{(n+q)\times (n+m)}$$ that we have obtained in the proof of Theorem 6.11 fulfils $$\textrm{rk}_{\mathbb{K}(z)} \left (z\tilde E - \tilde A\right )=n+q$$ by Proposition 6.1, since \begin{align} n+q=&\textrm{rk}_{\mathbb{K}(z)} \left[\begin{array}{@{}cc@{}} zE-A & -B \\ (z-1)K &(z-1)L\nonumber \end{array}\right]\\ =& \textrm{rk}_{\mathbb{K}(z)} \left[\begin{array}{@{}cc@{}} W & 0\\ 0 & I_{m}\nonumber \end{array}\right] \left[\begin{array}{@{}cc@{}} zE-A & -B \\ (z-1)K &(z-1)L \end{array}\right] \mathcal{T}_{F}\\ =& \textrm{rk}_{\mathbb{K}(z)} \left[\begin{array}{@{}ccc@{}} zI_{n_{1}}-A_{11} & 0 & -B_{1} \\ 0 & - I_{n_{2}} & -B_{2} \\ (z-1)K_{1} & (z-1)K_{2} &(z-1)L \end{array}\right]\\ =& \textrm{rk}_{\mathbb{K}(z)} \left[\begin{array}{@{}ccc@{}} zI_{n_{1}}-A_{11} & 0 & -B_{1} \\ 0 & - I_{n_{2}} & 0 \\ (z-1)K_{1} & 0 &(z-1)\left(L-K_{2}B_{2}\right)\nonumber \end{array}\right]. \end{align} (44) In particular, this means that the existence of solutions of (25) implies the existence of a deflating subspace of the palindromic pencil $$z\mathcal{A}^{\ast }-\mathcal{A}$$. (b) In the proof of Theorem 6.11 we have constructed a deflating subspace imYF as in (41) for the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form (17) from a solution (XF, KF, LF) of the Lur’e equation (25). From here we can construct a deflating subspace imY for the original system by using (38). By Lemma 6.7 it is justified to set $$ W^{-*}XW^{-1}=X_{F}:= \left[\begin{array}{@{}cc@{}} X_{11} & 0\\ 0 &0 \end{array}\right]. $$ Thus, we have \begin{align*} Y:=\!U_{F}Y_{F}\hat{V} \mathcal T_{F}^{-1} &=\left[ \begin{array}{@{}c|cc@{} } W^{\ast} & 0 & 0\\ \hline 0 & T & 0 \\ 0 & FT & I_{m} \end{array}\right] \left[ \begin{array}{@{}ccc@{} } X_{11}\left(A_{11}-I_{n_{1}}\right) & 0 & X_{11}B_{1} \\ 0 & -I_{n_{2}} & -B_{2}\\ \hline I_{n_{1}} & 0 & 0 \\ 0 & 0 & -B_{2} \\ 0 & 0 & I_{m} \end{array}\right] \mathcal T_{F}^{-1} \\ &= \left[ \begin{array}{@{}c|cc@{} } W^{\ast} & 0 & 0\\ \hline 0 & T & 0 \\ 0 & FT & I_{m} \end{array}\right] \!\left[ \begin{array}{@{}c|cc@{} } -X_{F} + \left(I_{n} - E_{F}\right) & 0 & 0\\ \hline 0 & I_{n} & 0 \\ 0 & 0 & I_{m} \end{array} \right] \!\left[ \begin{array}{@{}ccc@{} } I_{n_{1}} -A_{11} & 0 & -B_{1} \\ 0 & -I_{n_{2}} & -B_{2}\\ \hline I_{n_{1}} & 0 & 0 \\ 0 & 0 & -B_{2} \\ 0 & 0 & I_{m} \end{array}\right]\! \mathcal T_{F}^{-1} \\ &= \left[\begin{array}{@{}cc@{}} X(A-E) + G_{1} & XB + G_{2} \\ V_{1} & V_{2} \end{array}\right], \end{align*} where \begin{align} \operatorname{im} \left[\begin{array}{@{}cc@{}} G_{1} & G_{2} \end{array}\right] = \operatorname{im} W^{\ast} \left[\begin{array}{@{}cc@{}} 0 & 0 \\ 0 & I_{n_{2}} \end{array}\right] W \left[\begin{array}{@{}cc@{}} E-A & -B \end{array}\right] \subseteq \ker E^{\ast}, \end{align} (45) \begin{align} \left[\begin{array}{@{}cc@{}} V_{1} & V_{2} \end{array}\right]:=\mathcal T_{F} V_{F} \mathcal T_{F}^{-1}, \end{align} (46) and $$ V_{F}:= \left[\begin{array}{@{}ccc@{}} I_{n_{1}} & 0 & 0 \\ 0 & 0 & -B_{2} \\ 0 & 0 & I_{m} \end{array}\right] $$ spans the system space $$\mathcal{V}_{(E_F,A_F,B_F)}$$, see Proposition 2.11(a). Altogether, this leads to $$\left (z\mathcal{A}^{\ast }-\mathcal{A}\right ) Y=Z\left (z\check E-\check A\right )$$, where $$Z=U_{F}^{*}Z_{F}$$ and $$z\check E-\check A:=\left (z\tilde E - \tilde A\right )\mathcal T_{F}^{-1}$$. Example 6.13 (Example 2.7 revisited) Consider the system $$(E, A, B, Q, S, R)$$ as in (4) and Example 3.7. Note that since n3 = 0 in (5), the system $$(E, A, B)$$ is I-controllable according to Table 2. We have seen in Example 6.3 that $$ \left(X_{s},\, K_{s},\, L_{s}\right) = \left(\sqrt 3,\,{\sqrt2},\, -\frac{\sqrt3+1}{\sqrt2}\right) $$ is a solution of the Lur’e equation (27) corresponding to the EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s) $$ as in (20). By Lemma 6.7 we obtain that $$ X_{F}= \left[\begin{array}{@{}cc@{}} \sqrt3 & 0\\ 0 & 0 \end{array}\right], \quad K_{F}= \left[\begin{array}{@{}cc@{}} {\sqrt2} & 0 \end{array}\right], \quad L_{F} = -\frac{\sqrt3+1}{\sqrt2} $$ solves the Lur’e equation of the system in feedback equivalence form. Therefore, by Lemma 6.6 we see that \begin{align*} &X=W^{\ast}X_{F}W= \left[\begin{array}{@{}cc@{}} 1 & -1\\ 1 & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} \sqrt{3} & 0\\ 0 & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} 1 & 1\\ -1 & 0 \end{array}\right]= \left[\begin{array}{@{}cc@{}} \sqrt{3} & \sqrt{3}\\ \sqrt{3} & \sqrt{3} \end{array}\right],\\ &K=K_{F}T^{-1} \left[\begin{array}{@{}cc@{}} {\sqrt2} & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} 0 & 1 \\ 1 & -1 \end{array}\right] = \left[\begin{array}{@{}cc@{}} 0 &{\sqrt2} \end{array}\right], \quad L= -\frac{\sqrt3+1}{\sqrt2} \end{align*} solves the Lur’e equation (25) corresponding to the original system. Thus, according to Remark 6.12 the matrix $$Y\in \mathbb{K}^{5\times 3}$$ defined by \begin{align} Y= \left[\begin{array}{@{}cc@{}} X(A-E) + G_{1} & XB + G_{2} \\ V_{1} & V_{2} \end{array}\right] =\left[ \begin{array}{@{}cc|c@{}} 0 + 1 & 0 - 1 & -\sqrt{3} +1 \\ 0 + 0 & 0 + 0 & -\sqrt{3} + 0\\ \hline 0 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \end{align} (47) is a basis matrix of the deflating subspace of the palindromic pencil in (23). As in the EDE case we can show a similar statement for BVD pencils as in (12). Theorem 6.14 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be I-controllable. Further, let the corresponding BVD pencil $$z\mathcal{E}-\mathcal{A}$$ as in (12) be given. In addition, let $$q=\textrm{rk}_{\mathbb{K}(z)} \Phi (z)$$. Then, the following are equivalent: (a) There exists a solution $$(X,\,K,\,L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ of the Lur’e equation (25). (b) It holds that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all ω ∈ [0, 2π) such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. Furthermore, there exist matrices $$Y_{\mu },\,Y_{x}\in \mathbb{K}^{n\times (n+m)},\,Y_{u}\in \mathbb{K}^{m\times (n+m)}$$ and $$Z_{\mu },\,Z_{x}\in \mathbb{K}^{n\times (n+q)},\,Z_{u}\in \mathbb{K}^{m\times (n+q)}$$ such that for $$ Y=\left[\begin{array}{@{}c@{}} Y_{\mu}\\ Y_{x}\\ Y_{u} \end{array}\right],\qquad Z=\left[\begin{array}{@{}c@{}} Z_{\mu}\\ Z_{x}\\ Z_{u} \end{array}\right] $$ the following holds: (i) The matrix $$ \left[\begin{array}{@{}cc@{}} E & 0 \end{array}\right] \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right] $$ has rank n1. (ii) The space $$\mathcal Y= \operatorname{im} Y$$ is of dimension n + m and $$\mathcal{E}^{e}$$-neutral, where $$ \mathcal{E}^{e}:= \left[\begin{array}{@{}ccc@{}} 0 & -E & 0\\ E^{\ast}& 0 & 0\\ 0 & 0 & 0 \end{array}\right]. $$ (iii) It holds that $$ \mathcal{V}_{(E,A,B)}= \operatorname{im} \left[\begin{array}{@{}c@{}} Y_{x}\\ Y_{u} \end{array}\right]. $$ (iv) There exist $$\tilde E,\,\tilde A \in \mathbb{K}^{(n+q)\times (n+m)}$$ such that $$\left (z\mathcal{E}-\mathcal{A}\right )Y=Z\left (z\tilde E - \tilde A\right )$$. Proof. See the article by Bankmann (2016, Theorem 5.14). Remark 6.15 Let an I-controllable system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and let $$z\mathcal{E}-\mathcal{A}$$ be the BVD pencil as in (12). Further, assume that there exists a solution (X, K, L) of the Lur’e equation (25). (a) The matrix pencil $$\left (z\tilde E - \tilde A\right )\in \mathbb{K}[z]^{(n+q)\times (n+m)}$$ we obtain in the proof of Theorem 6.14 fulfils $$\textrm{rk}_{\mathbb{K}(z)} \left (z\tilde E - \tilde A\right )=n+q$$, see (44). In particular, this means that the existence of solutions of (25) implies the existence of a deflating subspace of the BVD pencil $$z\mathcal{E}-\mathcal{A}$$. (b) In the proof of Theorem 6.14 we have constructed a deflating subspace YF for the weighted system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form (17) from a solution (XF, KF, LF) of the Lur’e equation (25). From here we can construct a deflating subspace Y for the original system. By Lemma 6.7 it is justified to set $$ W^{-*}XW^{-1}=X_{F}:= \left[\begin{array}{@{}cc@{}} X_{11} & 0\\ 0 &0 \end{array}\right]. $$ Thus, we have $$ Y:=U_{F}Y_{F} \hat{V} \mathcal T_{F}^{-1} =\left[ \begin{array}{@{}c|cc@{} } W^{\ast} & 0 & 0\\ \hline 0 & T & 0 \\ 0 & FT & I_{m} \end{array}\right] \left[ \begin{array}{@{}ccc@{} } -X_{11} & 0 & 0 \\ 0 & -I_{n_{2}} & -B_{2}\\ \hline I_{n_{1}} & 0 & 0 \\ 0 & 0 & -B_{2} \\ 0 & 0 & I_{m} \end{array}\right] \mathcal T_{F}^{-1} = \left[\begin{array}{@{}cc@{}} -XE + G_{1} & G_{2} \\ V_{1} & V_{2} \end{array}\right], $$ where V1, V2 are as in (46) and \begin{align} \operatorname{im} \left[\begin{array}{@{}cc@{}} G_{1} & G_{2} \end{array}\right] = \operatorname{im} W^{\ast} \left[\begin{array}{@{}ccc@{}} 0 & 0 & 0\\ 0 & -I_{n_{2}} & -B_{2} \end{array}\right]\mathcal T_{F}^{-1} \subseteq \ker E^{\ast}. \end{align} (48) Altogether, this leads to $$\left (z\mathcal{E}-\mathcal{A}\right ) Y=Z\left (z\check E-\check A\right )$$, where $$Z=U_{F}^{*}Z_{F}$$ and $$z\check E-\check A:=\left (z\tilde E - \tilde A\right )\mathcal T_{F}^{-1}$$. Remark 6.16 A major difference between Theorem 6.11 and Theorem 6.14 or Theorem 6.4 and Theorem 6.5 is that in the BVD case we do not need the artificial assumption $$ \operatorname{rk} \left[\begin{array}{@{}cc@{}} E -A & -B \end{array}\right]=n, $$ or equivalently $$ \operatorname{rk} \left[\begin{array}{@{}cc@{}} I_{n_{1}} -A_{11} & -B_{1} \end{array}\right]=n_{1}, $$ i.e. controllability at one. If the system $$(E, A, B)$$ is obtained by discretization with the implicit Euler method, we see that in the limiting case h → 0 this corresponds to $$ \operatorname{rk} \lim\limits_{h\to0} \left[\begin{array}{@{}cc@{}} I_{n_{1}} -hA_{11} & -hB_{1} \end{array}\right] = \operatorname{rk} \left[\begin{array}{@{}cc@{}} I_{n_{1}} & 0 \end{array}\right] =n_{1}, $$ which is trivially fulfilled. Therefore, for sufficiently small h we may assume validity of this assumption. 7. Application to optimal control In this section we discuss the structure of solutions of the discrete-time optimal control problem (9) corresponding to the system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ based on the results from the previous section. First we show relations between so-called stabilizing solutions of the Lur’e equation (25) and feasibility of the optimal control problem. Then, we show characterizations for the existence and uniqueness of the optimal control. 7.1. Stabilizing solutions In this subsection we state several discrete-time versions of results from the article by Ilchmann & Reis (2017). If not explicitly stated otherwise, these results can be proven analogously, i.e. by using the same algebraic transformations and the identical properties of the solution operators. Definition 7.1 (Stabilizing solution) Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given and assume that a solution $$(X,\, K,\, L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{p\times n}\times \mathbb{K}^{p\times m}$$ of the corresponding Lur’e equation (25) exists. Then, (X, K, L) is also called stabilizing solution if in addition it holds \begin{align} \operatorname{rk} \left[\begin{array}{@{}cc@{}} \lambda E-A & -B \\ K &L \end{array}\right] = n+q \end{align} (49) for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq 1 $$. The following is an adaptation of Ilchmann & Reis (2017, Proposition 6.4(b)). Proposition 7.2 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ be I-controllable. If for every ε > 0 and every $$x^{0}\in \mathbb{K}^{n}$$ there exists a $$(x,u)\in \mathfrak{B}_{(E,A,B)} \cap \left (\ell ^{2}\left (\mathbb{K}^{n}\right ) \times \ell ^{2}\left (\mathbb{K}^{m}\right )\right )$$ such that Ex0 = Ex0 and $$\left \| Kx+ Lu\right \|_{\ell ^{2}} < \varepsilon ,$$ then we have \begin{align} \operatorname{rk} \left[\begin{array}{@{}ccc@{}} E & A & B\\ 0 & K & L \end{array}\right]= \operatorname{rk} \left[\begin{array}{@{}cc@{}} \lambda E - A & -B\\ K & L \end{array}\right] \end{align} (50) for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert>1 $$. Proof. The major part of the proof is completely analogous to the continuous-time case, since only algebraic operations and the linearity of the shift operator is used. The basic idea is that the relation (50) is equivalent to the fact that the Kronecker canonical form of the matrix pencil $$ \left[\begin{array}{@{}cc@{}} z E - A & -B\\ K & L \end{array}\right] $$ consists only of blocks of type K1 with $$\vert\lambda\vert \leq1 $$ and blocks of type K4 of size 1 × 0 (Ilchmann & Reis, 2017, Remark 2.8(b)). Also note that due to the block structure of the KCF we have that for every ε > 0 and every $$v^{\, j,0}\in \mathbb{C}^{k_{j}}$$ and corresponding block Kj(z) = zFj − Gj in the KCF there exists $$v^{\, j}\in \ell _{2}\left (\mathbb{C}^{k_{j}}\right )$$ such that $$ F_{j}{v_{0}^{\, j}} = F_{j}v^{\, j,0},\quad \lim_{i\to\infty}F_{j}{v^{\, j}_{i}}=0,\quad \left\|F_{j}\sigma v^{\, j} - G_{j}v^{\, j}\right\|_{\ell_{2}}<\varepsilon. $$ Here we show that all blocks of type K1 fulfil $$\vert\lambda\vert\leq 1 $$. To this end we thus proof the following fact: Let a block of the KCF of the form $$zF - G := K(z) = z I_{k} - \left (\lambda I_{k} + N_{k}\right ) \in \mathbb{C}[z]^{k\times k}$$ be given. If for any $$v^{0} \in \mathbb{C}^{k}$$ and any ε > 0 there exists a $$v\in \ell ^{2}\left (\mathbb{C}^{k}\right )$$ such that v0 = v0 and $$\left \|\sigma v - G v\right \|_{\ell_2} < \varepsilon $$, then $$\vert\lambda\vert \leq 1 $$. For the sake of a contradiction, assume that $$\vert\lambda\vert>1 $$. Then, the eigenvalue $$\frac 1\lambda $$ of G−1 fulfils $$\frac 1{|\lambda |}<1$$ and thus there exists positive definite P such that $$ P - G^{\ast}PG = - I_{k}. $$ Set $$\tilde{\varepsilon } = \left (v^{0}\right )^{\ast }Pv^{0}>0$$ and $$\varepsilon := \big \|\left (P+P^{2}\right )^{\frac 12} G^{-1}\big \|_{2}^{-2}\tilde{\varepsilon }$$. Then, for any $$v^{0}\in \mathbb{C}^{k}\setminus \{0\}$$ and w := σv − Gv with $$\|w\|^{2}_{\ell _{2}}<\varepsilon $$ one finds that \begin{align*} v_{0}^{\ast}Pv_{0}-v_{j}^{\ast}Pv_{j} &= -\sum_{k=0}^{j-1}{\sigma\left(v_{k}^{\ast}Pv_{k}\right) - v_{k}^{\ast}Pv_{k}} = -\sum_{k=0}^{j-1}{({Gv_{k} + w_{k})}^{\ast}P{(Gv_{k} + w_{k})} - v_{k}^{\ast}Pv_{k}}\\ &=-\sum_{k=0}^{j-1}{ \left(\begin{array}{@{}c@{}} v_{k}\\ w_{k} \end{array}\right)^{\ast} \left[\begin{array}{@{}cc@{}} G^{\ast}PG-P & G^{\ast}P\\ PG & P \end{array}\right] \left(\begin{array}{@{}c@{}} v_{k}\\ w_{k} \end{array}\right) }\\ & = - \sum_{k=0}^{j-1}{ \left(\begin{array}{@{}c@{}} v_{k}+ (I+P)G^{-1}w_{k}\\ G^{-1}w_{k} \end{array}\right)^{\ast} \left[\begin{array}{@{}cc@{}} I & 0\\ 0 & -P-P^{2} \end{array}\right] \left(\begin{array}{@{}c@{}} v_{k}+ (I+P)G^{-1}w_{k}\\ G^{-1}w_{k} \end{array}\right) }\\&\le \big\| (P+P^{2} )^{\frac{1}{2}} G^{-1}\big\|_{2}^{2}\|w\|^{2}_{\ell^{2}}<\tilde{\varepsilon} \end{align*} and thus in the limit case $$\tilde{\varepsilon } = \lim _{j\to \infty } v_{0}^{\ast }Pv_{0} - v_{j}^{\ast }Pv_{j}< \tilde{\varepsilon }$$. Lemma 7.3 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given. Further, assume that $$(X,\, K,\, L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{p\times n}\times \mathbb{K}^{p\times m}$$ fulfils (25) with \begin{align} \operatorname{rk} \left[\begin{array}{@{}cc@{}} K & L \end{array}\right] = p. \end{align} (51) If for every ε > 0 and every $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ there exists a $$(x,u)\in \mathfrak{B}_{(E,A,B)} \cap \left (\ell ^{2}\left (\mathbb{K}^{n}\right ) \times \ell ^{2}\left (\mathbb{K}^{m}\right )\right )$$ such that Ex0 = Ex0 and $$\left \| Kx+ Lu\right \|_{\ell ^{2}} < \varepsilon ,$$ then there exists a stabilizing solution of the Lur’e equation (49). Proof. By the rank condition (51) we can use the analogous version of “⇒” of Ilchmann & Reis (2017, Proposition 6.5) for Y = Ip, where Proposition 7.2 is used. Lemma 7.4 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be stabilizable. Further, assume that $$(X,\, K,\, L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ is a stabilizing solution of the Lur’e equation (25). Then, any given sequence $$v\in \ell ^{2}\left (\mathbb{K}^{q}\right )$$ can be approximated arbitrarily well by y = Kx + Lu and some $$(x,u)\in \mathfrak{B}_{(E,A,B)} \cap \left (\ell _{2}(\mathbb{K}^{n})\times \ell _{2}(\mathbb{K}^{m})\right )$$. In other words, for every ε > 0 and every $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ there exists $$(x,u)\in \mathfrak{B}_{(E,A,B)} \cap \left (\ell _{2}(\mathbb{K}^{n})\times \ell _{2}(\mathbb{K}^{m})\right )$$ such that Ex0 = Ex0 and $$ \left\|v - Kx- Lu\right\|_{\ell^{2}} < \varepsilon. $$ Proof. Since (X, K, L) is a stabilizing solution one can easily show that $$ \operatorname{rk} \left[\begin{array}{@{}ccc@{}} E & A &B\\ 0 & K &L \end{array}\right]= n+ q. $$ Then, the result is obtained by using the discrete-time version of Ilchmann & Reis (2017, Proposition 6.4(a)). The proof is tedious but analogous and relies on a discrete-time version of Ilchmann & Reis (2017, Theorem 5.1), which relates the algebraic characterization (49) to the fact that arbitrary sequences $$v\in \ell ^{2}(\mathbb{K})$$ can be approximated arbitrary well by y = Kx + Lu and given $$(x,u)\in \mathfrak{B}_{(E,A,B)} \cap \left (\ell _{2}\left (\mathbb{K}^{n}\right )\times \ell _{2}\left (\mathbb{K}^{m}\right )\right )$$. The argumentation uses the Hardy space $$\mathcal{H}_{2}^{q}$$ of analytic functions $$\textbf{G} : \left \{ \lambda \in \mathbb{C} \,:\, \left | \lambda \right |> 1 \right \} \to \mathbb{C}^{q}$$ with $$\int _{0}^{2\pi } \left \|\textbf{G}\left (\textrm{e}^{\textrm{i} \omega }\right )\right \|_{\textrm{F}}^{2} \ \textrm{d}\omega < \infty $$ and revolves around the fact that the multiplication operator mapping the Z-transform of the input to the Z-transform of the output has dense range in $$\mathcal{H}_{2}^{q}$$. 7.2. Feasibility We characterize feasibility and structure of the optimal control problem with the existence of a stabilizing solution of the Lur’e equation. First we show that the existence of a stabilizing solution implies feasibility and an explicit characterization of the optimal value function $$\mathcal{W}_{+}(Ex^0)$$ for given $$x^{0}\in \mathcal{W}_{(E,A,B)} $$. Theorem 7.5 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given with no uncontrollable modes on the unit circle and assume there exists a stabilizing solution $$(X,\, K,\, L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ of the Lur’e equation (49). Then, the optimal control problem is feasible, i.e. $$\mathcal{W}_{+}(Ex^0) \in \mathbb{R}$$ for all $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and $$\mathcal{W}_{+}(Ex^0) = \left (x^{0}\right )^{\ast }E^{\ast }XEx^{0}$$. Proof. First, since $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ has no uncontrollable modes on the unit circle and (X, K, L) is a stabilizing solution we obtain that $$ \operatorname{rk} \left[\begin{array}{@{}cc@{}} \lambda E - A & -B \end{array}\right] + q = \operatorname{rk} \left[\begin{array}{@{}cc@{}} \lambda E-A & -B \\ K &L \end{array}\right] = n+q $$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$. Thus, $$(E, A, B, Q, S, R)$$ is stabilizable. Let $$x^{0}\in \mathcal{W}_{(E,A,B)} $$. Then, we have $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ with Ex0 = Ex0 and $$\lim _{j\to \infty }Ex_{j}=0$$. By the definition of the system space $$\mathcal{V}_{(E,A,B)} \subseteq \mathbb{K}^{n+m}$$, we obtain $$({x_{j}^{*},u_{j}^{*}})^{*}\in \mathcal{V}_{(E,A,B)} $$ for all $$j\in \mathbb{N}_{0}$$. Thus, for $$j_{2}\geq j_{1} $$ we have that \begin{equation}\begin{aligned} x_{j_{2}}^{\ast}E^{\ast}XEx_{j_{2}} - x_{j_{1}}^{\ast}E^{\ast}XEx_{j_{1}} &= \sum_{k=j_{1}}^{j_{2}-1}{({Ax_{k} + Bu_{k})}^{\ast}X{\left(Ax_{k} + Bu_{k}\right)} - x_{k}^{\ast}E^{\ast}XEx_{k}}\\ &=\sum_{k=j_{1}}^{j_{2}-1}{ \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right)^{\ast} \left[\begin{array}{@{}cc@{}} A^{\ast}XA-E^{\ast}XE & A^{\ast}XB\\ B^{\ast}XA & B^{\ast}XB \end{array}\right] \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right) } \ge - \sum_{k=j_{1}}^{j_{2}-1}{ \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right)^{\ast} \left[\begin{array}{@{}cc@{}} Q & S\\ S^{\ast} & R \end{array}\right] \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right)}. \end{aligned}\end{equation} (52) For $$j_{1}=0,\,j_{2}\to \infty $$ we thus obtain for the objective function $$\mathcal{J}(x,u) $$ that \begin{equation*}x_{0}^{\ast}E^{\ast}XEx_{0}\le \mathcal{J}(x,u) \end{equation*} and thus \begin{align} x_{0}^{\ast}E^{\ast}XEx_{0}\le \mathcal{W}_{+}(Ex^0) < \infty. \end{align} (53) Furthermore, since (X, K, L) is a stabilizing solution of the Lur’e equation (25), for every $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and $$(x,u)\in \mathfrak{B}_{(E,A,B)} \cap \left (\ell ^{2}(\mathbb{K}^{n}) \times \ell ^{2}(\mathbb{K}^{m})\right )$$ with Ex0 = Ex0 we obtain in (52) that \begin{align*} -x_{0}^{\ast}E^{\ast}XEx_{0} =&\sum_{k=0}^{\infty}{ \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right)^{\ast} \left( \mathcal M (X) - \left[\begin{array}{@{}cc@{}} Q & S\\ S^{\ast} & R \end{array}\right] \right) \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right) }\\ =&\sum_{k=0}^{\infty}{ \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right)^{\ast} \left[\begin{array}{@{}c@{}} K^{\ast}\\ L^{\ast} \end{array}\right] \left[\begin{array}{@{}cc@{}} K & L \end{array}\right] \left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right) } - \mathcal{J}(x,u) \end{align*} and thus \begin{align} x_{0}^{\ast}E^{\ast}XEx_{0} + \left\|Kx + Lu\right\|_{\ell^{2}}^{2} = \mathcal{J}(x,u). \end{align} (54) Since (X, K, L) is a stabilizing solution, using Lemma 7.4 for v = 0 we further conclude that $$ \big(x^{0}\big)^{\ast}E^{\ast}XEx^{0} = \mathcal{W}_{+}(Ex^0) $$ for all $$x^{0}\in \mathcal{W}_{(E,A,B)} $$. Next we show that the opposite implication is also true, i.e. that feasibility implies the existence of a stabilizing solution of the Lur’e equation. Theorem 7.6 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given with no uncontrollable modes on the unit circle. Assume that the optimal control problem is feasible, i.e. $$\mathcal{W}_+ \left(Ex^{0}\right ) \in \mathbb{R}$$. Then, there exists a stabilizing solution of the Lur’e equation. Proof. First we have to show that $$\mathcal{W}_+ \left(Ex^{0}\right )= \left (Ex^{0}\right )^{\ast }XEx^{0}$$ for some Hermitian $$X\in \mathbb{K}^{n\times n}$$. This can be done in an analogous way as in the study Voigt (2015, Theorem 3.8.3). With similar steps as in the study by (Voigt (2015, Theorem 3.8.3) we can also show that X solves the KYP inequality (16). It remains to show that X also induces a stabilizing solution of the Lur’e equation. Since X solves the KYP inequality there exist $$\tilde K \in \mathbb{K}^{p\times n}$$ and $$\tilde L \in \mathbb{K}^{p\times m}$$ such that (25) and (51) hold. Thus, by (54) we obtain that for every $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and ε > 0 there exists $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that $$\big \|\tilde Kx + \tilde Lu\big \|_{\ell ^{2}}<\varepsilon $$. Then, by Lemma 7.3 we also obtain a stabilizing solution of the Lur’e equation. 7.3. Existence and uniqueness of optimal controls In this section we discuss conditions for the existence and uniqueness of optimal controls. Based on the considerations of the previous subsections we pose these conditions in terms of the zero dynamics $${\mathfrak{ZD}}_{(E,A,B,K,L)}$$ and the pencil $$\left[{{zE - A}\atop{K}}\quad {{-B}\atop{L}}\right]$$. Theorem 7.7 Let $$(E,A,B,Q,S,R) \in \Sigma _{m,n}^{w}(\mathbb{K})$$ be I-controllable and assume that $$(X,K,L) \in \mathbb{K}^{n \times n} \times \mathbb{K}^{q \times n} \times \mathbb{K}^{q \times m}$$ is a stabilizing solution of the Lur’e equation (25). Then, the following statements are satisfied: (a) For every $$x^{0} \in \mathbb{K}^{n}$$ there exists a trajectory $$(x,u) \in \mathfrak{B}_{(E,\,\, A,\,\,B)}$$ with Ex0 = Ex0 such that $$\mathcal{W}_{+}\big(Ex^{0}\big) = \mathcal{J}(x,u)$$ if and only if $${\mathfrak{ZD}}_{(E,A,B,K,L)}$$ is strongly stabilizable. (b) For every $$x^{0} \in \mathbb{K}^{n}$$ there exists a unique trajectory $$(x,u) \in \mathfrak{B}_{(E,\,\,A,\,\,B)}$$ with Ex0 = Ex0 such that $$\mathcal{W}_{+}\big(Ex^{0}\big) = \mathcal{J}(x,u)$$ if and only if $${\mathfrak{ZD}}_{(E,A,B,K,L)}$$ is strongly asymptotically stable. Proof. (a) In view of (54), we see that for every $$x^{0} \in \mathbb{K}^{n}$$ there exists a trajectory $$(x,u) \in \mathfrak{B}_{(E,A,B)}$$ with Ex0 = Ex0 such that $$\mathcal{W}_{+}\left (Ex^{0}\right ) = \mathcal{J}(x,u)$$, if and only for each $$x^{0} \in \mathbb{K}^{n}$$ there exists a $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,K,L)}\left (x^{0}\right )$$ with $$\lim _{j \to \infty } Ex_{j} = 0$$. Using Proposition 2.15(b), this is equivalent to $${\mathfrak{ZD}}_{(E,A,B,K,L)}$$ being strongly stabilizable. (b) For every $$x^{0} \in \mathbb{K}^{n}$$ there exists a unique trajectory $$(x,u) \in \mathfrak{B}_{(E,\,\,A,\,\,B)}$$ with Ex0 = Ex0 such that $$\mathcal{W}_{+}\big(Ex^{0}\big) = \mathcal{J}(x,u)$$, if and only for each $$x^{0} \in \mathbb{K}^{n}$$ there exists a unique $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,K,L)}\left (x^{0}\right )$$ with $$\lim _{j \to \infty } Ex_{j} = 0$$. With Proposition 2.15(d), this is equivalent to $${\mathfrak{ZD}}_{(E,A,B,K,L)}$$ being strongly asymptotically stable. Using the results of Proposition 2.15, we directly obtain the following corollary: Corollary 7.8 Let $$(E,A,B,Q,S,R) \in \Sigma _{m,n}^{w}(\mathbb{K})$$ be I-controllable and assume that $$(X,K,L) \in \mathbb{K}^{n \times n} \times \mathbb{K}^{q \times n} \times \mathbb{K}^{q \times m}$$ is a stabilizing solution of the Lur’e equation (25). Define $$\mathcal{R}(z) := \left [{{zE-A}\atop{K}} \quad {{-B} \atop {L}}\right ] \in \mathbb{K}(z)^{(n+q) \times (n+m)}$$. Then, the following statements are satisfied: (a) For every $$x^{0} \in \mathbb{K}^{n}$$ there exists a trajectory $$(x,u) \in \mathfrak{B}_{(E,\,\,A,\,\,B)}$$ with Ex0 = Ex0 such that $$\mathcal{W}_{+}\left (Ex^{0}\right ) = \mathcal{J}(x,u)$$, if and only if $$ \operatorname{rk} \mathcal{R}(\lambda ) = n+q$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert \lambda\vert \geq 1 $$ and the index of $$\mathcal{R}(z)$$ is at most one. (b) For every $$x^{0} \in \mathbb{K}^{n}$$ there exists a unique trajectory $$(x,u) \in \mathfrak{B}_{(E,\,\,A,\,\,B)}$$ with Ex0 = Ex0 such that $$\mathcal{W}_{+}\left (Ex^{0}\right ) = \mathcal{J}(x,u)$$, if and only if $$ \operatorname{rk} \mathcal{R}(\lambda ) = n+m$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert \lambda\vert \geq 1$$ and the index of $$\mathcal{R}(z)$$ is at most one. Existence and uniqueness of optimal controls can also be read of the PKCF of the palindromic matrix pencil $$z \mathcal{A}^{\ast } - \mathcal{A}$$ in (14). For this, one would need to analyse the spectral properties of the matrix pencil $$\left [{{zE-A}\atop{(z-1)K}} \quad {{-B} \atop {(z-1)L}}\right ]$$ in (26) corresponding to a stabilizing solution of the Lur’e equation and the structure of the deflating subspaces of individual blocks of the PKCF in detail as in the thesis by Voigt (2015). For brevity of the article we leave out this result here. 7.4. Application to palindromic and BVD matrix pencils Let us now discuss implications of the aforementioned results for the structure of optimal control with respect to the associated palindromic and BVD matrix pencils. Thus, assume that $$(X,\, K,\, L)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{p\times m}$$ is a stabilizing solution of the Lur’e equation (25). If $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ is given, then $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ with Ex0 = Ex0 and $$\lim \nolimits _{j\rightarrow \infty } Ex_{j}= 0$$ is an optimal control if and only if $$\|Kx + Lu\|_{\ell ^{2}}=0$$. If this is the case, then $$({x,u})$$ fulfils $$ \left[\begin{array}{@{}cc@{}} E & 0 \\ 0 & 0 \end{array}\right] \left[\begin{array}{@{}c@{}} x_{j+1}\\ u_{j+1} \end{array}\right]= \left[\begin{array}{@{}cc@{}} A & B \\ K & L \end{array}\right] \left[\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right],\quad Ex_{0}=Ex^{0},\quad \lim\limits_{j\to\infty} Ex_{j}=0. $$ For an I-controllable system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) in feedback equivalence form with corresponding transformation matrices W, T, F we set $$ X_{F}= W^{-*}X W^{-1}= \left[\begin{array}{@{}cc@{}} X_{11} & X_{12}\\ X_{21} & X_{22} \end{array}\right]. $$ Since $$ E^{\ast}XE =T^{-*} \left[\begin{array}{@{}cc@{}} I_{n_{1}} & 0\\ 0 & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} X_{11} & X_{12}\\ X_{12}^{\ast} & X_{22} \end{array}\right] \left[\begin{array}{@{}cc@{}} I_{n_{1}} & 0\\ 0 & 0 \end{array}\right] T^{-1} = T^{-*} \left[\begin{array}{@{}cc@{}} I_{n_{1}} & 0\\ 0 & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} X_{11} & 0\\ 0 & 0 \end{array}\right] \left[\begin{array}{@{}cc@{}} I_{n_{1}} & 0\\ 0 & 0 \end{array}\right] T^{-1}, $$ by Lemma 6.7 we can without loss of generality set X12 = 0 and X22 = 0. In addition, further assuming that $$ \operatorname{rk} \left [E-A \quad B\right ]=n,$$ from Theorem 6.14 we obtain a deflating subspace $$Y\in \mathbb{K}[z]^{(2n+m)\times (n+m)}$$ of the BVD pencil $$z\mathcal{E}-\mathcal{A}$$ as in (12), i.e. we have $$Z\in \mathbb{K}^{(2n+m)\times (n+q)}$$ and a matrix pencil $$z\check E-\check A\in \mathbb{K}[z]^{(n+q)\times (n+m)}$$ such that $$\left (z\mathcal{E}-\mathcal{A} \right )Y=Z\left (z\check E-\check A\right )$$. It can be constructed as in Remark 6.15(b). Inserting $$\sigma $$ for z leads to \begin{align} \left[ \begin{array}{@{}c|cc@{} } 0 & \sigma E-A & -B\\ \hline \sigma A^{\ast}-E^{\ast} & -Q & -S \\ \sigma B^{\ast} & -S^{\ast} & -R \end{array}\right]& \left[ \begin{array}{@{}c|c@{} } -XE + G_{1} & G_{2}\\ \hline V_{1} & V_{2} \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right)\nonumber\\ &=Z \left[ \begin{array}{@{}ccc@{} } \sigma I_{n_{1}}-A_{11} & 0 & -B_{1} \\ K_{1} & 0 & L-K_{2}B_{2} \\ -Q_{12}^{\ast} &- \sigma I_{n_{2}} & -\sigma B_{2} +Q_{22}B_{2}-S_{2} \end{array}\right] \mathcal T_{F}^{-1} \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right), \end{align} (55) where $$ \operatorname{im} \left [\,G_{1} \enspace G_{2}\,\right ]\subseteq \ker E^{\ast }$$, see (48), $$ \left[\begin{array}{@{}cc@{}} V_{1} & V_{2} \end{array}\right]:=\mathcal T_{F} V_{F} \mathcal T_{F}^{-1}, $$ and $$ V_{F}:= \left[\begin{array}{@{}ccc@{}} I_{n_{1}} & 0 & 0 \\ 0 & 0 & -B_{2} \\ 0 & 0 & I_{m} \end{array}\right]. $$ Since $$({x_{j},u_{j}})\in \mathcal{V}_{(E,A,B)} $$, it follows with Proposition 2.11(a) that $$ \mathcal T_{F}^{-1} \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right) = \left(\begin{array}{@{}c@{}} x_{1,j}\\ -B_{2}u_{j}\\ u_{j} \end{array}\right) $$ for some $$x_{1,j}\in \mathbb{K}^{n_{1}}$$. Then, by Lemma 6.7 and Lemma 6.6 we have that \begin{align} \left[\begin{array}{@{}cc@{}} K_{1} & L-K_{2}B_{2} \end{array}\right] \left(\begin{array}{@{}c@{}} x_{1,j} \\ u_{j} \end{array}\right) = \left[\begin{array}{@{}cc@{}} K_{F} & L_{F} \end{array}\right]\mathcal{T}_{F}^{-1} \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right) = \left[\begin{array}{@{}cc@{}} K & L \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right)=0. \end{align} (56) In addition, from (19) we obtain $$ \left[\begin{array}{@{}ccc@{}} Q_{12}^{\ast} & Q_{22} & S_{2} \end{array}\right]=K_{2}^{\ast} \left[\begin{array}{@{}ccc@{}} K_{1} & K_{2} & L \end{array}\right] $$ and thus \begin{equation}\begin{aligned} \left[\begin{array}{@{}cc@{}} Q_{12}^{\ast} & S_{2} -Q_{22}B_{2} \end{array}\right] \left(\begin{array}{@{}c@{}} x_{1,j} \\ u_{j} \end{array}\right) &= \left[\begin{array}{ccc} Q_{12}^{\ast} & Q_{22} & S_{2} \end{array}\right] \mathcal{T}_{F}^{-1} \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right) =K_{2}^{\ast} \left[\begin{array}{@{}ccc@{}} K_{1} & K_{2} & L \end{array}\right] \mathcal{T}_{F}^{-1} \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right)\\ &=K_{2}^{\ast} \left[\begin{array}{cc} K & L \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right)=0. \end{aligned}\end{equation} (57) Thus, by equations (56) and (57) the right-hand side of (55) is zero. Furthermore, by Proposition 2.11(d) we have that $$ V \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right) = \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right). $$ Set $$ \mu_{j} := \left[\begin{array}{@{}cc@{}} -XE + G_{1} & G_{2} \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right). $$ Thus, $$ \lim\limits_{j\to\infty}{E^{\ast}\mu_{j}} =\lim\limits_{j\to\infty}{-E^{\ast}XEx_{j}}=0, $$ and hence, $$\left (\mu _{j}\right )_{j}$$ is part of a solution of the boundary value problem $$ \left[\begin{array}{@{}ccc@{}} 0 & E & 0 \\ A^{\ast} & 0 & 0 \\ B^{\ast} & 0 & 0 \end{array}\right] \sigma \left(\begin{array}{@{}c@{}} \mu\\ x\\ u \end{array}\right) = \left[\begin{array}{@{}ccc@{}} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\right] \left(\begin{array}{@{}c@{}} \mu\\ x\\ u \end{array}\right),\qquad Ex_{0} = E x^{0}, \quad \lim\limits_{j\to\infty}{E^{\ast}\mu_{j}}=0. $$ Moreover, for an I-controllable system $$(E, A, B, Q, S, R)$$ we can take the same approach for a deflating subspace $$Y\in \mathbb{K}[z]^{(2n+m)\times (n+m)}$$ of the palindromic pencil $$z\mathcal{A}^{\ast }-\mathcal{A}$$ as in (14) obtained in Theorem 6.11. There we have $$Z\in \mathbb{K}^{(2n+m)\times (n+q)}$$ and a matrix pencil $$z\check E-\check A\in \mathbb{K}[z]^{(n+q)\times (n+m)}$$ such that $$\left (z\mathcal{A}^{\ast }-\mathcal{A}\right ) Y=Z\left (z\check E-\check A\right )$$. It can be constructed as in Remark 6.12(b). Inserting $$\sigma $$ for z leads to \begin{multline}\left[ \begin{array}{@{}c|cc@{} } 0 & \sigma E-A & -B\\ \hline \sigma A^{\ast}-E^{\ast} & (\sigma-1)Q & (\sigma-1)S \\ \sigma B^{\ast} & (\sigma-1)S^{\ast} & (\sigma-1)R \end{array}\right]\left[ \begin{array}{@{}c|c@{} } -X(E-A) +G_{1} & -XB + G_{2}\\ \hline V_{1} & V_{2} \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right)\\ =Z \left[ \begin{array}{@{}ccc@{} } \sigma I_{n_{1}}-A_{11} & 0 & -B_{1} \\ (\sigma -1)K_{1} & 0 & (\sigma -1)\left(L-K_{2}B_{2}\right) \\ (\sigma -1)Q_{12}^{\ast} & -\sigma I_{n_{2}} & -\sigma B_{2} +(\sigma -1) \left(S_{2} - Q_{22}B_{2} \right) \end{array}\right] \mathcal T_{F}^{-1} \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right), \end{multline} (58) where $$ \operatorname{im} \left [G_{1} \enspace G_{2}\right ]\subseteq \ker E^{\ast }$$, see (45). Again by equations (56) and (57) the right-hand side of (58) is 0. Set \begin{align} m_{j} := \left[\begin{array}{@{}cc@{}} X(A - E) + G_{1} & XB + G_{2} \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right). \end{align} (59) Thus, \begin{align*} \sum_{k=0}^{\infty}{E^{\ast}m_{k}}=& \sum_{k=0}^{\infty}{-E^{\ast}XEx_{k} + E^{\ast}X \left[\begin{array}{@{}cc@{}} A & B \end{array}\right] \!\left(\begin{array}{@{}c@{}} x_{k}\\ u_{k} \end{array}\right) }\\ =& \sum_{k=0}^{\infty}{-E^{\ast}XE\left(x_{k} - x_{k+1}\right)}\\ =&-E^{\ast}XEx_{0} = E^{\ast} \mu_{0} \end{align*} and hence, $$\left (m_{j}\right )_{j}$$ is part of a solution of the boundary value problem \begin{equation} \left[\begin{array}{@{}ccc@{}} 0 & E & 0 \\ A^{\ast} & Q & S \\ B^{\ast} & S^{\ast} & R \end{array}\right] \sigma \left(\begin{array}{@{}c@{}} m\\ x\\ u \end{array}\right) = \left[\begin{array}{@{}ccc@{}} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\right] \left(\begin{array}{@{}c@{}} m\\ x\\ u \end{array}\right),\quad Ex_{0} = E x^{0}, \quad \sum_{k=0}^{\infty}{E^{\ast}m_{k}}=E^{\ast}\mu_{0}. \end{equation} (60) Example 7.9 (Example 2.7 revisited) Consider the system $$(E, A, B, Q, S, R)$$ as in (4) and Example 3.7. In Example 6.13 we have seen that $$ (X,\,K,\,L)=\left( \left[\begin{array}{@{}cc@{}} \sqrt{3} & \sqrt{3}\\ \sqrt{3} & \sqrt{3} \end{array}\right],\, \left[\begin{array}{@{}cc@{}} 0 &{\sqrt2} \end{array}\right],\, -\frac{\sqrt3+1}{\sqrt2} \right) $$ is a solution of the Lur’e equation (25). We have $$ E^{\ast}XE= \left[\begin{array}{@{}cc@{}} 0 & 0\\ 0 & \sqrt3 \end{array}\right] $$ and thus by (53) for every $$ x^{0}= \left(\begin{array}{@{}c@{}} {x^{0}_{1}}\\[2pt] {x^{0}_{2}} \end{array}\right) \in\mathcal{W}_{(E,A,B)} $$ the optimal value $$\mathcal{W}_{+}(Ex^0) $$ is bounded from below by $$\sqrt 3 \,\left |{x^{0}_{2}}\right|{}\!^{2}$$. Indeed, setting $$ u_{j} = \frac{2}{\sqrt3 +1}\left(1-\frac{2}{\sqrt3 +1}\right)^{j} {x^{0}_{2}}, $$ we obtain that $$ x_{j} = \left(\begin{array}{c} 1-\frac{2}{\sqrt3 +1}\\ 1 \end{array}\right) \left(1-\frac{2}{\sqrt3 +1}\right)^{j} {x^{0}_{2}} $$ solves the system given by (4) with $$ \mathcal{J}(x,u) = \sum_{j=0}^{\infty}{ \left\|x_{j}\right\|^{2} + \left\|u_{j}\right\|^{2} } =\big|{x_{2}^{0}}\big|{}\!^{2}\frac{12-6\sqrt3}{1-\left(1-\frac{2}{\sqrt3 +1}\right)^{2}}=\big|{x_{2}^{0}}\big|{}\!^{2}\sqrt3, $$ i.e. (x, u) is an optimal control fulfilling Ex0 = Ex0 and $$\lim \limits _{j\to \infty }Ex_{j}=0$$. In particular, from (47) we obtain that $$ m_{j}= \left[\begin{array}{@{}ccc@{}} 1 & -1 & -\sqrt3 +1\\ 0 & 0 & -\sqrt3 \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right) = -\sqrt{3} \frac{2}{\sqrt3 +1} \left(\begin{array}{@{}c@{}} 1\\ 1 \end{array}\right) \left(1-\frac{2}{\sqrt3 +1}\right)^{j} {x^{0}_{2}} $$ fulfils the boundary value problem (60), where $$ \mu_{0}= -\sqrt3 \left(\begin{array}{@{}c@{}} 1\\ 1 \end{array}\right) {x_{2}^{0}}. $$ 8. Conclusions and outlook We have discussed several problems arising in the discrete-time linear-quadratic optimal control problem and we have seen their relations to the results that have been obtained in the continuous-time setting. In Section 3 we have discussed an extension of the KYP inequality for standard difference equations to the case of implicit difference equations. The characterizations are analogous to what was obtained in the studies by Reis et al. (2015) and Voigt (2015) in the continuous-time case. Nonetheless, some more technical difficulties had to be tackled. For an analogous relaxation of the controllability assumption to sign-controllability we would need the discrete-time analogue of Clements et al. (1997, Theorem 6.1). In Section 4 we further relate the spectral properties of the palindromic pencil associated to the discrete-time optimal control problem (9) to the positivity of the Popov function on the unit circle. To this end, we introduced the notion of quasi-Hermitian matrices which allows for a generalization of the concept of inertia. In Section 6 we introduced Lur’e equations for explicit as well as for IDEs. We have shown that solvability of these equations is equivalent to the existence of certain deflating subspaces of the BVD and palindromic pencil arising in the discrete-time control problem (9). In the palindromic case we needed the additional assumption that the given system is controllable at the eigenvalue one, which can always be achieved for discrete-time systems originating from discretization. It is an open question whether this condition can be dropped if the latter is not the case. In Section 7 we have seen how we can use these results to characterize feasibility of the optimal control problem as well as existence and uniqueness of optimal controls. Furthermore, we have shown how the deflating subspaces are related to the solutions of the related two-point boundary value problems. Funding The first author has been partially supported by the European Research Council through the Advanced Grant ‘Modeling, Simulation and Control of Multi-Physics Systems’ (MODSIMCONMP). 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Lecture Notes in Mathematics, vol. 973 . Berlin : Springer , pp. 58 -- 73 . Voigt , M. ( 2015 ) On linear-quadratic optimal control and robustness of differential-algebraic systems . Also as Dissertation, Fakultät für Mathematik, Otto-von-Guericke-Universität Magdeburg . Berlin : Logos . ISBN 978–3-8325–4118-7 . © The Author(s) 2018. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

On linear-quadratic optimal control of implicit difference equations

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Abstract

Abstract In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur’e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur’e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context. 1. Introduction In this article we revisit the discrete-time linear-quadratic optimal control problem that is minimizing a quadratic cost functional given by $$ \sum_{j=0}^{\infty} \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right)^{\ast} \left[\begin{array}{@{}cc@{}} Q & S\\ S^{\ast} & R \end{array}\right] \left(\begin{array}{@{}c@{}} x_{j}\\ u_{j} \end{array}\right) $$ subject to the implicit difference equation (IDE) \begin{align} E\sigma x_{j} = Ax_{j} + Bu_{j}, \end{align} (1) with the initial condition Ex0 = Ex0 and the stabilization condition $$\lim _{j \to \infty } Ex_{j} = 0$$. Here $$\sigma $$ denotes the shift operator, i.e. $$\sigma x_{j} = x_{j+1}$$. Moreover, $$\left (x_{j}\right )_{j}\in (\mathbb{K}^{n})^{\mathbb{N}_{0}}$$ is the state sequence, and $$\left (u_{j}\right )_{j}\in (\mathbb{K}^{m})^{\mathbb{N}_{0}}$$ is the input sequence. this work we will further assume that the matrix pencil $$zE-A\in \mathbb{K}[z]^{n\times n}$$ is regular, i.e. $${\det (zE-A)\not \equiv 0}$$. Such discrete-time systems often appear during the time-discretization (Brenan et al., 1996) or discrete-time lifting (Kahane et al., 1999) of continuous-time differential-algebraic equations, but many problems can also be directly modelled as implicit difference equations (Luenberger & Arbel, 1977; Pearson et al., 1988). There is a large body of work concerning the linear-quadratic optimal control problem for differential-algebraic equations, see, e.g. the studies by Mehrmann (1991), Lancaster & Rodman (1995), Backes (2006), Kurina & März (2004) and Reis et al. (2015), just to mention a few. So far, the discrete-time optimal control problem has only been discussed in a few works, most of which treat the case E = In. However, several technical assumptions are usually made. In the study by Laub (1979), the case of an invertible A with Q$$\succeq $$ 0 and R ≻ 0 is discussed. The invertibility is needed to form a symplectic matrix that is associated with a discrete-time algebraic Riccati equation \begin{align} A^{\ast}XA-X - (A^{\ast}XB + S)(B^{\ast}XB+R)^{-1} (B^{\ast}XA + S^{\ast}) + Q = 0, \quad X=X^{\ast} \end{align} (2) and the necessary optimality conditions. The invertibility condition is relaxed in the studies by Pappas et al. (1980) and Laub (1991) where instead of the symplectic matrix a symplectic matrix pencil is considered. Another difficulty arises if R is not invertible. Then, also the symplectic pencil cannot be formed and one has to turn to an extended symplectic pencil (Ionescu & Weiss, 1992), which is essentially what we will later call the BVD matrix pencil (here, BVD is an acronym for Boundary Value problem for the optimal control of Discrete systems). However, for the analysis it is still assumed that this pencil is regular. The relation of the optimal control problem to a certain linear matrix inequality (the Kalman–Yakubovich–Popov (KYP) inequality) and so-called Popov functions is discussed in the article by Stoorvogel & Saberi (1998). Here, a generalized algebraic Riccati equation is considered, where B*XB + R is not assumed to be invertible. It is shown that the solutions of this equation fulfil a certain rank-minimization property of the associated linear matrix inequality which, in contrast to the continuous-time case, do not need to be solutions of the algebraic Riccati equation (2). The case of optimal control problems for implicit difference equations, i.e. the case where E might be singular, has only been briefly considered in the literature. In the article by Bender & Laub (1987), a discrete-time algebraic Riccati equation similar to (2) is derived by transforming the system into SVD coordinates and modifying the cost functional accordingly. This analysis needs an index-one condition on the system to ensure the solvability of the optimality system. The monograph Mehrmann (1991) treats the problem numerically, i.e. structure-preserving algorithms for symplectic matrix pencils are devised. The goal of this work is a full theoretical analysis of the infinite time horizon linear-quadratic optimal control problem for implicit difference equations. In contrast to most other works, we do not impose any definiteness conditions on the cost functional nor the index of the system. Also, our notion of rank-minimality turns out to be more general than in the article by Stoorvogel & Saberi (1998). The results obtained in this paper are motivated by recent achievements for the continuous-time case (Reis, 2011; Reis et al., 2015; Voigt, 2015). This paper is structured as follows: in Section 2 we recap basic matrix and control theoretic notations and results. In Section 3 we introduce a variant of the KYP inequality for implicit difference equations given by $$ \mathcal M (P):= \left[\begin{array}{@{}cc@{}} A^{\ast}PA - E^{\ast}PE + Q &A^{\ast}PB + S \\ B^{\ast}PA + S^{\ast} &B^{\ast}PB + R \end{array}\right] \succeq{\mathcal{V}}_{(E,A,B)} 0,\qquad P=P^{\ast}, $$ a discrete-time version of the inequality introduced in the article by Reis et al. (2015), where $$\succeq _{\mathcal{V}_{(E,A,B)}}$$ denotes an inequality projected on a certain subspace $$\mathcal{V}_{(E,A,B)} $$, i.e. $$V^{\ast }\mathcal M (P)V \succeq 0$$ holds for any basis matrix V of $$\mathcal{V}_{(E,A,B)} $$. We show statements which relate the solvability of this inequality to the non-negativity of the Popov function on the unit circle, a certain rational matrix function defined by $$ \Phi(z):= \left[\begin{array}{@{}c@{}} \left(zE-A\right)^{-1}B \\ I_{m} \end{array}\right]^{\sim} \left[\begin{array}{@{}cc@{}} Q & S \\ S^{\ast} & R \end{array}\right] \left[\begin{array}{@{}c@{}} (zE-A)^{-1}B \\ I_{m} \end{array}\right] \in\mathbb{K}(z)^{m\times m}, $$ where $$G^{\sim }(z):=G\left (\overline{z}^{-1}\right )^{\ast }$$ for a rational matrix $$G(z)\in \mathbb{K}(z)^{n\times n}$$. In Sections 4 and 5 we introduce the notion of inertia for palindromic matrix pencils evaluated on the unit circle and provide spectral characterizations regarding positivity of the Popov function, similar to the characterizations which were obtained in the studies by Reis (2011) and Voigt (2015) for even matrix pencils in the continuous-time case. In Section 6 we investigate the Lur’e equation for the discrete-time optimal control problem which is a generalization of the algebraic Riccati equation (2). This means that we seek solution triples $$(X,\,K,\,L)\in \mathbb{K}^{n\times n}\times \mathbb{K}^{q\times n}\times \mathbb{K}^{q\times m}$$ fulfilling $$ \left[\begin{array}{@{}cc@{}} A^{\ast}XA-E^{\ast}XE + Q & A^{\ast}XB + S \\ B^{\ast}XA + S^{\ast} & B^{\ast}XB + R \end{array}\right] ={\!}_{\mathcal{V}_{(E,A,B)}} \left[\begin{array}{@{}c@{}} K^{\ast}\\ L^{\ast} \end{array}\right] \left[\begin{array}{@{}cc@{}} K & L \end{array}\right], \quad X=X^{\ast}, $$ where $$q:=\mathrm{rk}_{\mathbb{K}_(z)} \Phi(z)$$. We show that solvability of this equation can be related to the existence of certain deflating subspaces of a palindromic matrix pencil of the form $$ z \left[\begin{array}{@{}ccc@{}} 0 & E & 0 \\ A^{\ast} & Q & S \\ B^{\ast} & S^{\ast} & R \end{array}\right] - \left[\begin{array}{@{}ccc@{}} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\right]\in\mathbb{K}[z]^{(2n+m)\times (2n+m)}. $$ The explicit construction of these deflating subspaces turns out to be much more demanding compared to the continuous-time setting. Furthermore, solutions of the Lur’e equation can be obtained from these subspaces. Due to the symmetry of the above pencil, robust and efficient structure-exploiting numerical methods (Byers et al., 2009; Schröder, 2008) can be used. Finally, in Section 7 we apply these results to the optimal control problem. Here, we consider the stabilizing solutions of the Lur’e equations which can be used to construct optimal controls (in case of existence) and to determine the optimal value. In particular, we show that under some weak conditions, the existence of stabilizing solutions is equivalent to the feasibility of the optimal control problem. We further characterize existence and uniqueness of optimal controls in terms of the zero dynamics of the closed-loop system. Finally, we discuss how the deflating subspaces of the palindromic and BVD pencils appear in the solution of the corresponding boundary value problems. Note that parts of these results are obtained in the master's thesis by Bankmann (2016). Nomenclature $$\mathbb{N}$$ = {1, 2, …}; set of natural numbers $$\mathbb{N}_{0}$$$$=\mathbb{N}\cup \{0\}$$ $$\mathbb{R}$$ field of real numbers $$\mathbb{R}^{+}$$ set of positive real numbers $$\mathbb{R}^{+}_{0}$$ set of non-negative real numbers $$\mathbb{C}$$ field of complex numbers $$\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$$ $$\mathbb{K}[z]$$ ring of polynomials with coefficients in $$\mathbb{K}$$ $$\mathbb{K}(z)$$ field of rational functions that can be expressed as fraction of elements of $$\mathbb{K}[z]$$ $${e_{i}^{k}}$$i-th unit vector in $$\mathbb{K}^{k}$$ $$K^{\mathbb{N}_{0}}$$ set of all sequences $$x=\left (x_{j}\right )_{j}$$ whose components lie in the space K $$\mathcal{R}^{m \times n}$$ set of m by n matrices with entries in a ring $$\mathcal{R}$$ $$\det A$$ determinant of a matrix $$A\in \mathbb{K}^{n \times n}$$ A* conjugate transpose of a matrix $$A\in \mathbb{K}^{m\times n}$$ A+ Moore–Penrose pseudo inverse of a matrix $$A\in \mathbb{K}^{m\times n}$$ A−* conjugate transpose of the inverse of an invertible matrix $$A\in \mathbb{K}^{n \times n}$$ $$\textrm{rk}_{\mathbb{K}(z)} A(z)$$ rank of a rational matrix $$A(z)\in \mathbb{K}[z]^{m\times n}$$ G∼ (z) $$:=G\left (\overline{z}^{-1}\right )^{\ast }$$ for a rational matrix $$G(z)\in \mathbb{K}[z]^{n\times n}$$ ∥x∥2 two-norm of a vector $$x\in \mathbb{K}^{n}$$ $$\ell ^{2}\left (\mathbb{K}^{n}\right )$$ space of quadratic-summable sequences $$x\in (\mathbb{K}^{n})^{\mathbb{N}_{0}}$$, i.e. $$\sum _{k=0}^{\infty }{\left \| x_{j}\right \|_{2}}<\infty $$ $$\| x\|_{\ell ^{2}}$$$$=\left (\sum _{k=0}^{\infty }{\left \| x_{j}\right \|^{2}_{2}}\right )^{\frac 12}$$; ℓ2-norm of a sequence $$x\in \ell ^{2}(\mathbb{K}^{n})$$ $$\Sigma_{m,n}(\mathbb{K})$$ set of all $$(E,A,B)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}$$ with regular zE − A $$\Sigma_{m,n}^w(\mathbb{K})$$ set of all $$(E, A, B, Q, S, R)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{m\times m}$$, where $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ and Q and R are Hermitian $$\mathfrak{B}_{(E,A,B)}$$ set of all $$({x,u})$$ which solve the discrete-time implicit difference equation (1) $$\mathcal{V}_{(E,A,B)} $$ system space of $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$, see Def. 2.9 $$\mathcal{W}_{(E,A,B)} $$ set of all $$x^{0}\in \mathbb{K}^{n}$$ such that there exists $$(x,u)\in \mathfrak{B}_{(E,A,B)}$$ with Ex0 = Ex0 $${\mathfrak{ZD}}_{(E,A,B,C,D)}$$ set of all $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that $$Cx_{j} + Du_{j} =0,\,j\in \mathbb{N}_{0}$$ $$\mathcal W^{0}_{(E,A,B,C,D)}$$ set of all $$x^{0}\in \mathbb{K}^{n}$$ such that there exists $$({x,u})\in {\mathfrak{ZD}}_{(E,A,B,C,D)}$$ with Ex0 = Ex0 2. Mathematical preliminaries 2.1. Matrix pencils In this section we briefly discuss some basic notions of matrix pencils $$zE-A\in \mathbb{K}[z]^{m\times n}$$. Definition 2.1 (Equivalence of matrix pencils) Two matrix pencils $$zE_{1} - A_{1}, \, zE_{2} - A_{2} \in \mathbb{K}[z]^{m\times n}$$ are called equivalent if there exist invertible matrices $$W \in \mathbb{K}^{m\times m}$$ and $$T \in \mathbb{K}^{n \times n}$$ such that $$ zE_{2} - A_{2} = W \left(zE_{1} - A_{1}\right) T. $$ Each matrix pencil $$zE-A \in \mathbb{C}[z]^{m\times n}$$ can be transformed to Kronecker canonical form via equivalence transformations. This is made precise in the next theorem. Theorem 2.2 (Kronecker canonical form (KCF)) (Gantmacher, 1960) For every matrix pencil $$zE-A\in \mathbb{C}[z]^{m\times n}$$, there exist invertible matrices $$W\in \mathbb{C}^{m\times m}$$ and $$T\in \mathbb{C}^{n\times n}$$ such that $$ W\left(zE-A\right)T = \operatorname{diag} \left( {K}_{1}(z),\,\ldots,\,{K}_{l}(z)\right),\qquad l\in\mathbb{N}, $$ where each block Kj(z) is in one of the forms in Table 1 and $$ N_{k}= \left[\begin{array}{@{}cccc@{}} 0& 1 & & \\ & \ddots&\ddots &\\ & & \ddots&1\\ &&&0 \end{array}\right]\in \mathbb{C}^{k\times k},\, {K^{R}_{k}} = \left[\begin{array}{@{}ccc@{}} 1& &\\ 0&\ddots&\\ &\ddots &1\\ &&0 \end{array}\right]\in\mathbb{C}^{(k+1)\times k},\, {K^{L}_{k}} = \left[\begin{array}{@{}cccc@{}} 0& 1 & & \\ & \ddots&\ddots &\\ & &0&1 \end{array}\right]\in\mathbb{C}^{k\times (k+1)}. $$ The KCF is unique up to permutations of the blocks. Table 1. Blocks in Kronecker canonical form Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ View Large Table 1. Blocks in Kronecker canonical form Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ Type Size Kj(z) Parameters K1 k × k (z − λ)Ik − Nk $$ k\in \mathbb{N},\,\lambda \in \mathbb{C}$$ K2 k × k zNk − Ik $$ k\in \mathbb{N}$$ K3 k × (k + 1) $$z\left ({K_{k}^{R}}\right )^{T} - {K_{k}^{L}} $$ $$ k\in \mathbb{N}_{0}$$ K4 (k + 1) × k $$z{K_{k}^{R}} - \left ({K_{k}^{L}}\right )^{T} $$ $$ k\in \mathbb{N}_{0}$$ View Large Equivalent matrix pencils share the same spectral structure which can be read off the KCF. Here, blocks of type K1 and K2 correspond to finite eigenvalues and infinite eigenvalues, respectively. Blocks of these types and combinations of them are regular. Blocks of types K3 and K4 are rectangular and thus not regular. Note that we allow for blocks of type K3 or K4 to have zero rows or zero columns, respectively. Such blocks represent a zero row or zero column, in the KCF of zE − A, respectively. Based on the KCF we define the index of the pencil $$zE-A \in \mathbb{C}[z]^{m\times n}$$ as the size k of the largest block of type K2 or K4 in its KCF (Berger & Reis, 2013). When characterizing the eigenstructure of matrices $$A\in \mathbb{K}^{n \times n}$$, often invariant subspaces are involved, i.e. subspaces $$\mathcal V \subseteq \mathbb{K}^{n}$$ such that $$A\mathcal V \subseteq \mathcal V$$. The generalization of invariant subspaces to matrix pencils $$zE-A\in \mathbb{K}[z]^{m\times n}$$ are so-called deflating subspaces. Here, we are using a general definition which is also suitable for singular matrix pencils, see the studies by Van Dooren (1983) and Voigt (2015). Definition 2.3 (Basis matrix, deflating subspaces) Let $$zE-A\in \mathbb{K}[z]^{n\times n}$$ and some subspace $$\mathcal{Y}\subseteq \mathbb{K}^{n}$$ be given. (a) A matrix $$Y\in \mathbb{C}^{n\times k}$$ with full column rank such that $$\mathcal{Y} = \operatorname{im} Y$$ is called basis matrix of $$\mathcal{Y}$$. (b) If for a basis matrix $$Y\in \mathbb{C}^{n\times k}$$ of $$\mathcal{Y}$$ there exist $$W\in \mathbb{C}^{n\times p}$$ and $$z\hat{E}- \hat{A}\in \mathbb{C}[z]^{p\times n}$$ such that $$ \left(zE-A\right)V = W \left(z\hat{E} - \hat{A}\right) $$ and $$\textrm{rk}_{\mathbb{C}(z)} \left (z\hat{E}- \hat{A}\right ) = p$$, then $$\mathcal{Y}$$ is called deflating subspace of zE − A. Indeed, every invariant subspace $$\mathcal V \subseteq \mathbb{K}^{n}$$ of $$A\in \mathbb{K}^{n \times n}$$ with basis matrix $$V\in \mathbb{K}^{n\times k}$$ describes a deflating subspace for the associated matrix pencil zIn − A by setting W = V and $$z\hat{E} - \hat{A}= zI_{k} -\Lambda $$, where $$\Lambda \in \mathbb{K}^{k\times k}$$ fulfils AV = VΛ. An important property that deflating subspaces might have is E-neutrality. Definition 2.4 (E-neutrality) (Gohberg et al., 2006; Reis, 2011) Let $$E\in \mathbb{K}^{n \times n}$$ and some subspace $$\mathcal{Y}\subseteq \mathbb{K}^{n}$$ be given. Then, $$\mathcal{Y}$$ is called E-neutral if for all $$x,\,y\in \mathcal{Y}$$ it holds that x*Ey = 0. It is called maximally E-neutral if every proper superspace $$\mathcal{W}\supsetneq \mathcal{Y}$$ is not E-neutral. For a subspace $$\mathcal{Y}\subseteq \mathbb{K}^{n}$$ we can check E-neutrality by testing whether Y*EY = 0, where $$Y\in \mathbb{K}^{n\times k}$$ is given such that $$ \operatorname{im} Y = \mathcal{Y}$$. 2.2. Feedback equivalence Let $$\Sigma_{m,n}(\mathbb{K}) $$ denote the set containing all the system triples $$(E,A,B)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}$$ with regular zE − A, i.e. $$\det (zE-A)\neq 0$$. Later, we will also use the set $$\Sigma_{m,n}^w(\mathbb{K}) $$ containing all systems $$(E, A, B, Q, S, R)\in \mathbb{K}^{n \times n}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{n \times n}\times \mathbb{K}^{n\times m}\times \mathbb{K}^{m\times m}$$, where $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$ and Q and R are Hermitian. Furthermore, we call the space of all $$(\,{x,\, u}\,)\in (\mathbb{K}^{n})^{\mathbb{N}_{0}}\times (\mathbb{K}^{m})^{\mathbb{N}_{0}}$$ that solve the IDE (1) the behaviour of the system $$(E,A,B)$$. The behaviour is denoted by $$\mathfrak{B}_{(E,A,B)}$$. In this subsection we introduce an equivalence relation on the set $$\Sigma_{m,n}(\mathbb{K}) $$ which will be particularly useful in Sections 3 and 6. This subsection is mainly based on the article by Reis et al. (2015, Section 2.3). Definition 2.5 (Feedback equivalence) Two systems $$(E_{i},\,A_{i},\,B_{i})\in \Sigma_{m,n}(\mathbb{K}) ,\,i=1,2$$, are said to be feedback equivalent if there exist invertible matrices $$W,\,T\in \mathbb{K}^{n \times n}$$ and a feedback matrix $$F\in \mathbb{K}^{m\times n}$$ such that \begin{equation*} \big[zE_{2}-A_{2}\quad -B_{2}\big] = W \big[zE_{1}-A_{1}\quad -B_{1}\big]\,\mathcal{T}_{F}, \end{equation*} where $$ \mathcal{T}_{F} = \left[\begin{array}{@{}cc@{}} T & 0 \\ FT & I_{m} \end{array}\right]. $$ If this is the case we say that (E1, A1, B1) is feedback equivalent to (E2, A2, B2) via W and $$\mathcal{T}_{F}$$. Note that in the behaviour sense, i.e. looking at the system defined by $$z\mathcal{E}-\mathcal{A}$$, where $$ \mathcal{E} := [E\quad 0],\qquad \mathcal{A}:= [A\quad B], $$ feedback equivalence corresponds to strong equivalence as introduced in the book by Kunkel & Mehrmann (2006). In particular, this means that feedback equivalence is indeed an equivalence relation, (Kunkel & Mehrmann, 2006, Lemma 2.2.) Given such an equivalence relation, one is usually interested in some condensed form. The following result provides such a form. Theorem 2.6 (Feedback equivalence form) (Ilchmann & Reis, 2017, Proposition 2.12) Let the system $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$ be given. Then, $$(E,A,B)$$ is feedback equivalent to $$(E_{F},A_{F},B_{F})$$ via some W and $$\mathcal T_{F}$$, where \begin{align} [zE_{F} - A_{F}\quad -B_{F}]=\left[ \begin{array}{@{} ccc|c@{}} zI_{n_{1}}-A_{11} & 0 & 0 & -B_{1} \\ 0 & - I_{n_{2}} & zE_{23} & -B_{2} \\ 0 & 0 & zE_{33} - I_{n_{3}} & 0 \end{array}\right], \end{align} (3)$$n_{1}, n_{2}, n_{3}\in \mathbb{N}_{0}$$ and E33 is nilpotent. A similar form has also been achieved in the article by Byers et al. (1997, Theorem 4.1) via unitary transformations. Example 2.7 Consider the system given by \begin{align} E= \left[\begin{array}{@{}cc@{}} 0 & 0\\ 0 & 1 \end{array}\right],\quad A= \left[\begin{array}{@{}cc@{}} -1 & 1 \\ 1 & 0\\ \end{array}\right],\quad B= \left[\begin{array}{@{}c@{}} -1\\ 0 \end{array}\right]. \end{align} (4) We obtain that the system is feedback equivalent to \begin{align} [z E_{F} - A_{F}\quad -B_{F}]= W \left[z E- A\quad -B\right]\mathcal T_{F}=\left[\begin{array}{@{}ccc@{}} z -1 & 0 & 1\\ 0 & -1 & -1 \end{array}\right] \end{align} (5) via zero feedback, i.e. F = 0 and $$ W= \left[\begin{array}{@{}cc@{}} 1 & 1\\ -1 & 0 \end{array}\right], \quad\mathcal T_{F}= \left[\begin{array}{@{}ccc@{}} 1 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{array}\right]. $$ Thus, we have n1 = n2 = 1 and n3 = 0 in (3). Proposition 2.8 Let the system $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$ be feedback equivalent to the system $$(E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ in feedback equivalence form (3). Further, denote by $$ (I_{n_{1}},A_{11},B_{1})\in \Sigma_{m,n1}(\mathbb{K}) $$ the associated explicit difference equation (EDE) system. Then, for $$\lambda \in \mathbb{C}$$ we have $$\det \left (\lambda E_{F} -A_{F}\right )\neq 0$$ if and only if $$\det \left (\lambda I_{n_{1}} - A_{11}\right )\neq 0$$. Proof. Note that every nilpotent matrix $$E_{33}\in \mathbb{K}^{n_{3}\times n_{3}}$$ has only eigenvalues zero and thus $$\det \left (zE_{33}-I_{n_{3}}\right ) = (-1)^{n_{3}}$$. Then, the assertion follows immediately from the block-triangular structure of zEF − AF. 2.3. System space In this subsection we investigate properties of the solution space of the IDEs given by a system $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$. This section is based on the article by Reis et al. (2015, Chapter 3). Definition 2.9 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$. The smallest subspace $$\mathcal{V}_{(E,A,B)} \subseteq \mathbb{K}^{n+m}$$ such that $$ \left(\begin{array}{@{}c@{}} x_{j} \\ u_{j} \end{array}\right) \in \mathcal{V}_{(E,A,B)} $$ for all $$j\in \mathbb{N}_{0}$$ and for all $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ are called the system space of $$(E,A,B)$$. Lemma 2.10 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K}) $$. Further, assume that $$(E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is feedback equivalent to $$(E,A,B)$$ via W and $$\mathcal{T}_{F}$$. Then, the system spaces $$\mathcal{V}_{(E,A,B)} $$ and $$\mathcal{V}_{(E_F,A_F,B_F)}$$ of $$(E, A, B)$$ and $$(E_{F},A_{F},B_{F})$$, respectively, are related via $$ \mathcal{V}_{(E,A,B)} = \mathcal{T}_{F} \mathcal{V}_{(E_F,A_F,B_F)}. $$ Proof. The assertion has been shown in the article by Reis et al. (2015, Lemma 3.2). Proposition 2.11 Let $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ be given. Further, assume that $$(E_{F},A_{F},B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is feedback equivalent to $$(E, A, B)$$ via W and $$\mathcal{T}_{F}$$ such that $$(E_{F},A_{F},B_{F})$$ is in feedback equivalence form (3). Then, we have: (a) It holds that $$\mathcal{V}_{(E_F,A_F,B_F)} = \operatorname{im} V_{F}$$, where \begin{align} V_{F} := \left[\begin{array}{@{}cccc@{}} I_{n_{1}} & 0 & 0 & 0 \\ 0 & 0 & 0 & -B_{2}\\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & I_{m} \end{array}\right]\in\mathbb{K}^{(n+m)\times (n+m)}. \end{align} (6) (b) It holds that \begin{align} [A \ \ B] \left[\begin{array}{@{}c@{}} \left(zE-A\right)^{-1}B\\ I_{m} \end{array}\right] = z \ \left[E \ \ 0\right]\left[\begin{array}{@{}c@{}} (zE-A)^{-1}B\\ I_{m} \end{array}\right]. \end{align} (7) (c) For all $$\lambda \in \mathbb{C}$$ with $$\det (\lambda E - A)\neq 0 $$ it holds that $$ \operatorname{im} \left[\begin{array}{@{}c@{}} (\lambda E-A)^{-1}B\\ I_{m} \end{array}\right] \subseteq \mathcal{V}_{(E,A,B)} . $$ (d) Consider VF as in (a) and let $$V:= \mathcal T_{F} V_{F}$$. Then, $$ V_{F} \mathcal{V}_{(E_F,A_F,B_F)}= \mathcal{V}_{(E_F,A_F,B_F)} $$ and $$ V \mathcal T_{F}^{-1} \mathcal{V}_{(E,A,B)} = \mathcal{V}_{(E,A,B)}. $$ Proof. Assertion (a) is shown in the proof of Reis et al. (2015, Proposition 3.3). Assertion (c) is shown in the study by Reis et al. (2015, Lemma 3.5), where part (b) is obtained in the proof of Reis et al. (2015, Lemma 3.5). For part (d) see the study by Bankmann (2016, Proposition 2.29(d)). 2.4. Controllability and asymptotic stability Before we introduce the linear-quadratic optimal control problem, we first need to recap several concepts of controllability and asymptotic stability for the system given by $$(E, A, B)\in \Sigma_{m,n}(\mathbb{K})$$. These concepts are similar to the continuous-time case as in the studies by Dai (1989) and Bunse-Gerstner et al. (1999) and are discussed in, e.g. the studies by Dai (1989) and Stykel (2003). Definition 2.12 The system $$(E, A, B)\in \Sigma_{m,n}(\mathbb{K}) $$ or $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ is called (a) completely controllable (C-controllable) if for every initial point $$x^{0}\in \mathbb{K}^{n}$$ and every final point $$x^{\textrm{f}}\in \mathbb{K}^{n}$$ there exists a $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that x0 = x0 and $$x_{j_{\textrm{f}}}=x^{\textrm{f}}$$ at some timepoint $$j_{\textrm{f}}\in \mathbb{N}_{0}$$; (b) controllable on the reachable set (R-controllable) if for every initial point $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and every final point $$x_{\textrm{f}}\in \mathcal{W}_{(E,A,B)} $$ there exist $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0 and $$Ex_{j_{\textrm{f}}}=Ex^{\textrm{f}}$$ at some timepoint $$j_{\textrm{f}}\in \mathbb{N}_{0}$$; (c) I-controllable if for every initial point $$x^{0}\in \mathbb{K}^{n}$$ there exists $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0, i.e. $$\mathcal{W}_{(E,A,B)} =\mathbb{K}^{n}$$; (d) stabilizable if for every initial point $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ there exists a $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0 and $$\lim \nolimits_{j\to \infty } Ex_{j} = 0$$. In the case where E = In, the notions R-controllability and C-controllability coincide and thus for systems of the form $$(I_{n}, A, B)\in \Sigma_{m,n}(\mathbb{K})$$ we omit the prefix R or C and say that they are controllable. Table 2 shows well-known characterizations of the different controllability notions (Dai, 1989; Stykel, 2003; Berger, 2014). Table 2. Algebraic characterizations of controllability and stabilizability, where $$S_{\infty }$$ is a basis matrix of $$\ker E$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ View Large Table 2. Algebraic characterizations of controllability and stabilizability, where $$S_{\infty }$$ is a basis matrix of $$\ker E$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ Notion Algebraic characterization R-controllability rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ C-controllability rk[E B] = n, rk[λE−A B] = n, $$\lambda \in \mathbb{C}$$ I-controllability $$ \operatorname{rk} [E\,\ AS_{\infty }\ \, B] =n$$ Stabilizability rk[λE−A B] = n, $$\lambda \in \mathbb{C},\, |\lambda | \ge 1$$ View Large Eigenvalues $$\lambda \in \mathbb{C}$$ of $$zE-A\in \mathbb{K}[z]^{n\times n}$$ such that $$ \operatorname{rk} \left [\lambda E -A \quad B \right ] \neq n$$ destroy the controllability property. In this case we thus say that there is an uncontrollable mode at λ; otherwise we say that there is a controllable mode at λ. Lemma 2.13 Let the system $$(E, A, B)\in \Sigma_{m,n}(\mathbb{K})$$ be feedback equivalent to the system $$(E_{F}, A_{F}, B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ in feedback equivalence form (3) via W and $$\mathcal T_{F}$$. Furthermore, denote by $$(I_{n_{1}},A_{11},B_{1})\in \Sigma_{m,n1}(\mathbb{K})$$ the associated EDE system. Then, we have: (a) Let $$\lambda \in \mathbb{C}$$. Then, the system $$(E_{F},A_{F},B_{F})$$ has an uncontrollable mode at λ if and only if the system $$(I_{n_{1}}, A_{11}, B_{1})$$ has an uncontrollable mode at λ. (b) The system $$(E_{F},A_{F},B_{F})$$ is R-controllable if and only if $$(I_{n_{1}}, A_{11}, B_{1})$$ is controllable. (c) The system $$(E, A, B)$$ is I-controllable if and only if W and $$\mathcal T_{F}$$ can be chosen such that for $$(E_{F},A_{F},B_{F})$$ it holds that n3 = 0. Proof. Assertion (a) is shown in the study by Reis et al. (2015, Lemma 2.9(c)). Then, assertion (b) is an immediate consequence of (a), since by Table 2 the system $$(E_{F},A_{F},B_{F})$$ is R-controllable if and only if all $$\lambda \in \mathbb{C}$$ there are controllable modes of $$(E_{F},A_{F},B_{F})$$ at λ. Part (c) is shown in the study by Ilchmann & Reis (2017, Proposition 2.12). 2.5. Zero dynamics In this subsection we consider IDEs with an output of the form \begin{align} E\sigma x_{j} = Ax_{j} + Bu_{j}, \quad y_{j} = Cx_{j} + Du_{j}, \end{align} (8) where $$C \in \mathbb{K}^{q \times n}$$ and $$D \in \mathbb{K}^{q \times m}$$. The set of such systems with $$(E,A,B)\in \Sigma_{m,n}(\mathbb{K})$$ is denoted by $$\Sigma _{m,n,q}(\mathbb{K})$$ and we write $$(E,A,B,C,D) \in \Sigma _{m,n,q}(\mathbb{K})$$. The zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ of (8) simply consists of all $$(x,u) \in \mathfrak{B}_{(E,A,B)}$$ that result in a zero output, i.e. $$ {\mathfrak{ZD}}_{(E,A,B,C,D)} := \left\{ (x,u) \in \left(\mathbb{K}^{n}\right)^{\mathbb{N}_{0}} \times \left(\mathbb{K}^{m}\right)^{\mathbb{N}_{0}} \; \bigg| \; \left[\!\!\begin{array}{cc} E \sigma - A & -B \\ C & D \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{j} \\ u_{j} \end{array}\!\!\right) = 0,\, j = 0,\,1,\,2,\,\ldots \right\}\!. $$ The set of zero dynamics with ‘initial state’ $$x^{0} \in \mathbb{K}^{n}$$ is defined by $$ {\mathfrak{ZD}}_{(E,A,B,C,D)}\big(x^{0}\big) := \left\{ (x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} \; \Big| \; Ex_{0} = Ex^{0} \right\}. $$ The set of consistent initial shift variables for the zero dynamics is given by $$ \mathcal{W}^{0}_{(E,A,B,C,D)} := \left\{ x^{0} \in \mathbb{K}^{n} \; \Big| \; {\mathfrak{ZD}}_{(E,A,B,C,D)}(x^{0}) \neq \emptyset \right\}. $$ The following definition is an adaptation of the definition for continuous-time systems, see the studies by Voigt (2015) and Ilchmann & Reis (2017). Definition 2.14 The zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ with set of consistent initial shift variables $$\mathcal{W}^{0}_{(E,A,B,C,D)} $$ is called (a) stabilizable, if for all $$x^{0} \in \mathcal{W}^{0}_{(E,A,B,C,D)} $$, there exists an $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} \left (x^{0}\right )$$ such that $$\lim _{j \to \infty } \left (x_{j},u_{j}\right ) = 0$$; (b) asymptotically stable, if for all $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} $$ it holds $$\lim _{j \to \infty } \left (x_{j},u_{j}\right ) = 0$$; (c) strongly stabilizable, if it is stabilizable and $$\mathcal{W}^{0}_{(E,A,B,C,D)} = \mathbb{K}^{n}$$; (d) strongly asymptotically stable, if it is asymptotically stable and $$\mathcal{W}^{0}_{(E,A,B,C,D)} = \mathbb{K}^{n}$$. Proposition 2.15 Let $$(E,A,B,C,D) \in \Sigma _{m,n,q}(\mathbb{K})$$ be given and define $$\mathcal{R}(z) := \left[{{zE - A} \atop {C}}\quad {{-B} \atop {D}}\right] \in \mathbb{K}[z]^{(n+q) \times (n+m)}$$. Then, the zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ with the space of consistent shift variables $$\mathcal{W}^{0}_{(E,A,B,C,D)} $$ is (a) stabilizable, if and only if $$ \operatorname{rk}_{\mathbb{K}(z)} \mathcal{R}(z) = \operatorname{rk} \mathcal{R}(\lambda )$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$; (b) stabilizable, if and only if for all $$x^{0} \in \mathcal{W}^{0}_{(E,A,B,C,D)} $$ there exists a $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} \left (x^{0}\right )$$ such that $$\lim _{j \to \infty } Ex_{j} = 0$$; (c) asymptotically stable, if and only if $$ \operatorname{rk} \mathcal{R}(\lambda ) = n+m$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq 1 $$; (d) asymptotically stable, if and only if for all $$x^{0} \in \mathcal{W}^{0}_{(E,A,B,C,D)} $$ there exists a unique $$(x,u) \in {\mathfrak{ZD}}_{(E,A,B,C,D)} (x^{0})$$ such that $$\lim _{j \to \infty } Ex_{j} = 0$$; (e) strongly stabilizable, if and only if $$ \operatorname{rk}_{\mathbb{K}(z)} \mathcal{R}(z) = \operatorname{rk} \mathcal{R}(\lambda )$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$ and the index of $$\mathcal{R}(z)$$ is at most one; (f) strongly asymptotically stable, if and only if $$ \operatorname{rk} \mathcal{R}(\lambda ) = n+m$$ for all $$\lambda \in \mathbb{C}$$ with $$\vert\lambda\vert\geq1 $$ and the index of $$\mathcal{R}(z)$$ is at most one; Proof. The proof follows the lines of the proof of Ilchmann & Reis (2017, Prop. 4.3). It is analogously verified that the zero dynamics $${\mathfrak{ZD}}_{(E,A,B,C,D)} $$ is (a) stabilizable, if and only if all blocks of type K1 in the KCF of $$\mathcal{R}(z)$$ correspond to eigenvalues with |λ| < 1; (b) asymptotically stable, if and only if all blocks of type K1 in the KCF of $$\mathcal{R}(z)$$ correspond to eigenvalues with |λ| < 1 and the number of blocks of type K3 in the KCF of $$\mathcal{R}(z)$$ is zero. Together with the results in Ilchmann & Reis (2017, Rem. 2.5 (d), (e)), this shows statements (a)–(d). Moreover, it is checked that $$\mathcal{W}^{0}_{(E,A,B,C,D)} = \mathbb{K}^{n}$$ is equivalent to the condition that for every $$x^{0} \in \mathbb{K}^{n}$$ the IDE $$ \left[\!\!\begin{array}{cc} E & 0 \\ 0 & 0 \end{array}\!\!\right]\sigma \left(\!\!\begin{array}{c} x_{j} \\ u_{j} \end{array}\!\!\right) = \left[\!\!\begin{array}{cc} A & B \\ C & D \end{array}\!\!\right]\left(\!\!\begin{array}{c} x_{j} \\ u_{j} \end{array}\!\!\right), \quad Ex_{0} = Ex^{0} $$ has a solution. This is equivalent to $$\mathcal{R}(z)$$ being of index at most one (Berger, 2014) and shows statements (e) and (f). 2.6. Linear-quadratic optimal control One main goal of this work is to provide tools for analysing the discrete-time infinite horizon linear-quadratic control problem (Mehrmann, 1991; Kurina & März, 2004; Backes, 2006). It is given by the following: For $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ find $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0, $$\lim \limits _{j\rightarrow \infty } Ex_{j}= 0$$ and the objective function \begin{align} \mathcal{J}(x,u) := \sum_{j=0}^{\infty}{ \left(\!\!\begin{array}{c} x_{j}\\ u_{j} \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} Q & S\\ S^{\ast} & R \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{j}\\ u_{j} \end{array}\!\!\right) } \end{align} (9) is minimized. In other words, we are interested in the value of the functional $$\mathcal{W}_{+}(Ex^0) :E\mathcal{W}_{(E,A,B)} \rightarrow \mathbb{R}^{+}_{0}\cup \left \{\pm \infty \right \}$$ defined by $$ \mathcal{W}_{+}(Ex^0) := \inf \left\{ \mathcal{J}(x,u) \,\middle|\, (x,u)\in \mathfrak{B}_{(E,A,B)},\, Ex_{0}=Ex^{0},\,\lim\limits_{j\rightarrow\infty} Ex_{j}= 0 \right\}. $$ The problem is called feasible if $$\infty>\mathcal{W}_{+}(Ex^0) >-\infty $$. It is called solvable if the infimum is actually a minimum. Note that for $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ the existence of $$(x,u)\in \mathfrak{B}_{(E,A,B)} $$ such that Ex0 = Ex0 is guaranteed by the definition of $$\mathcal{W}_{(E,A,B)} $$. If further $$(E, A, B)$$ is stabilizable we can choose u such that in addition $$\lim \nolimits _{j\rightarrow \infty } Ex_{j}= 0$$, i.e. $$\mathcal{W}_{+}(Ex^0) < \infty $$. It can be easily seen that the objective function $$\mathcal{J}(x,u) $$ does not change if the system $$(E_{F}, A_{F}, B_{F})\in \Sigma_{m,n}(\mathbb{K})$$ is equivalent to $$(E, A, B)$$ via W and $$\mathcal{T}_{F}$$ and we use the modified weights $$ \left[\!\!\begin{array}{cc} Q_{F} & S_{F}\\ S_{F}^{\ast}& R_{F} \end{array}\!\!\right]:= \left[\!\!\begin{array}{cc} T^{\ast}\left(Q + F^{\ast}S^{\ast} + SF + F^{\ast}RF\right)T & T^{\ast}\left(S+F^{\ast}R\right)\\ \left(S^{\ast}+RF\right)T & R \end{array}\!\!\right]. $$ If we assume that the system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ is I-controllable and that \begin{align} \left[\!\!\begin{array}{cc} Q & S\\ S^{\ast}& R \end{array}\!\!\right]\succeq 0 \end{align} (10) then it is well-known that in this case solutions of the optimal control problem can be characterized via certain structured matrix pencils, see the studies by Mehrmann (1991), Mackey et al. (2006), Kunkel & Mehrmann (2008) and Byers et al. (2009). One main contribution of this work is that we actually can drop these assumptions. In the discrete-time case, applying Pontryagin’s maximum principle (Pontryagin et al., 1962; Mehrmann, 1991) leads to \begin{align} \left[\!\!\begin{array}{ccc} 0 & E & 0 \\ A^{\ast} & 0 & 0 \\ B^{\ast} & 0 & 0 \end{array}\!\!\right] \sigma \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right) = \left[\!\!\begin{array}{ccc} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\!\!\right] \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right),\quad Ex_{0} = E x^{0}, \quad \lim\limits_{j\to\infty}{E^{\ast}\mu_{j}}=0, \end{align} (11) where $$(x,u)\in \mathfrak{B}_{(E,A,B)}$$, $$x^{0}\in \mathcal{W}_{(E,A,B)} $$ and $$\mu \in{(\mathbb{K}^{n})^{\mathbb{N}_{0}}}$$ denote some Lagrange multipliers. This IDE can be analysed by means of the matrix pencil \begin{align} z\mathcal{E}-\mathcal{A} = \left[\!\!\begin{array}{ccc} 0 & zE-A & -B \\ zA^{\ast}-E^{\ast} & -Q & -S \\ zB^{\ast} & -S^{\ast} & -R \end{array}\!\!\right]\in\mathbb{K}[z]^{(2n+m)\times (2n+m)}, \end{align} (12) a so-called BVD-pencil. The structure of this pencil is not invariant under unitary transformations which leads to problems in the numerical treatment (Byers et al., 2009). In the studies by Schröder (2008) and Byers et al. (2009) it is shown how we can achieve a more structured version if we introduce new variables $$ m_{j}:=\mu_{j} - \mu_{j+1}. $$ This reformulation yields \begin{align} \left[\!\!\begin{array}{ccc} 0 & E & 0 \\ A^{\ast} & Q & S \\ B^{\ast} & S^{\ast} & R \end{array}\!\!\right] \sigma \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right) = \left[\!\!\begin{array}{ccc} 0 & A & B \\ E^{\ast} & Q & S \\ 0 & S^{\ast} & R \end{array}\!\!\right] \left(\!\!\begin{array}{c} \mu\\ x\\ u \end{array}\!\!\right),\quad Ex_{0} = E x^{0}, \quad \sum\limits_{j=0}^{\infty}{E^{\ast}m_{j}}=E^{\ast}\mu_{j} \end{align} (13) with the corresponding matrix pencil \begin{align} z\mathcal{E}-\mathcal{A} = z\mathcal{A}^{\ast}-\mathcal{A} \left[\!\!\begin{array}{ccc} 0 & zE-A & -B \\ zA^{\ast}-E^{\ast} & (z-1)Q & (z-1)S \\ zB^{\ast} & (z-1)S^{\ast} & (z-1)R \end{array}\!\!\right]\in\mathbb{K}[z]^{(2n+m)\times (2n+m)}. \end{align} (14) This pencil has the special property of being palindromic, i.e. $$\mathcal{E}=\mathcal{A}^{\ast }$$. This structure is preserved under congruence transformation and there exist numerically stable and structure-preserving methods for the computation of eigenvalues and deflating subspaces. In particular, simple eigenvalues on the unit circle stay on the unit circle (Schröder, 2008; Byers et al., 2009). In Section 4 we discuss properties of palindromic pencils in more detail. However, in the article by Mehrmann & Scholz (2014) it is shown that in an abstract Banach space setting the operator associated to the palindromic pencil (14) is not self-adjoint; in contrast to the operator associated to the so-called even pencil arising in continuous-time, see also the article by Kunkel et al. (2014). We show in Sections 6 and 7 that in analogy to the continuous-time case study by Reis et al. (2015), also in the discrete-time case, we can drop assumption (10) to obtain the necessary optimality conditions (11) and (13). 3. Kalman-Yakubovich-Popov lemma Consider the weighted system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ and corresponding system space $$\mathcal{V}_{(E,A,B)} $$. In this section we relate positive semi-definiteness on the unit circle of the Popov function—a specific rational matrix function—to the solvability of a certain matrix inequality, namely the inequality. We will see in Section 7 that positive semi-definiteness on the unit circle of the Popov function is necessary for feasibility of the optimal control problem (9). Definition 3.1 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given. Consider $$P=P^{\ast }\in \mathbb{K}^{n \times n}$$ and \begin{align} \mathcal{M}(P):= \left[\!\!\begin{array}{cc} A^{\ast}PA - E^{\ast}PE + Q & A^{\ast}PB + S \\ B^{\ast}PA + S^{\ast} & B^{\ast}PB + R \end{array}\!\!\right]. \end{align} (15) If $$\mathcal{M}(P)\succeq{}_{\mathcal{V}_{(E,A,B)}}0$$, then P is called solution of the discrete-time KYP inequality \begin{align} \mathcal{M}(P)\succeq{}_{\mathcal{V}_{(E,A,B)}} 0,\qquad P^{\ast}=P. \end{align} (16) Throughout this section we will make use of the system $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ being transformed to $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ via feedback equivalence, i.e. we have invertible $$W,\,T\in \mathbb{K}^{n \times n}$$ and a feedback matrix $$F\in \mathbb{K}^{m\times n}$$ such that \begin{align} E_{F}=&\, WET,\quad A_{F}=W(A+BF)T,\quad B_{F}=WB,\nonumber \\ Q_{F}=&\, T^{\ast}\left(Q+SF+F^{\ast}S^{\ast}+F^{\ast}RF\right)T,\quad S_{F}=T^{\ast}\left(S+F^{\ast}R\right),\quad R_{F}=R. \end{align} (17) These transformations will allow us to extract an EDE formulation from the IDE problem. The next results are crucial for the proof of the KYP lemma in the IDE case and are mainly adaptions of the corresponding results in Reis et al. (2015, Section 4). Lemma 3.2 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$. Then, we have $$ \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} A^{\ast}PA-E^{\ast}PE & A^{\ast}PB\\ B^{\ast}PA & B^{\ast}PB \end{array}\!\!\right] \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right] = 0. $$ Proof. By using (7) we obtain \begin{align*} &\left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} A^{\ast}PA-E^{\ast}PE & A^{\ast}PB\\ B^{\ast}PA & B^{\ast}PB \end{array}\!\!\right] \left[\!\! \begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right] \\ &\quad \overset{\hphantom{\eqref{eq:sysspace}}}= \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left( \left[\!\!\begin{array}{c} A^{\ast}\\ B^{\ast} \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} A & B \end{array}\!\!\right] - \left[\!\!\begin{array}{c} E^{\ast}\\ 0 \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} E & 0 \end{array}\!\!\right] \right) \left[ \begin{array}{@{}c@{}} (zE-A)^{-1}B \\ I_{m} \end{array} \right] \\ &\quad \overset{(7)}= \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left( \overline{z}^{-*} \left[\!\!\begin{array}{c} E^{\ast}\\ 0 \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} E & 0 \end{array}\!\!\right] z - \left[\!\!\begin{array}{c} E^{\ast}\\ 0 \end{array}\!\!\right] P \left[\!\!\begin{array}{cc} E & 0 \end{array}\!\!\right] \right) \left[\!\!\begin{array}{c} (zE-A)^{-1}B \\ I_{m} \end{array}\!\!\right] = 0. \end{align*} Lemma 3.3 Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ with corresponding system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ in feedback equivalence form as in (17) be given. Further, let $$P=P^{\ast }\in \mathbb{K}^{n \times n}$$ and set PF = W−*PW−1 and $$ \mathcal{T}_{F}= \left[\!\!\begin{array}{cc} T & 0 \\ FT& I_{m} \end{array}\!\!\right]. $$ Then, $$ \mathcal{M}_{F}(P_{F})= \mathcal{T}_{F}^{\ast } \mathcal{M}(P) \mathcal{T}_{F}, $$ where $$\mathcal{M}_{F}(P_{F})$$ is the matrix in (15) with respect to $$(E_{F}, A_{F}, Q_{F}, S_{F}, R_{F})$$. For the generalization of the KYP inequality to IDEs we first need to understand relations between the different Popov functions and KYP inequalities corresponding to systems $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ and $$ (E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) and how they are related to EDEs. If $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ is also in feedback equivalence form (3), then the associated EDE part is given by $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K})$$ which is defined by \begin{align} A_{s}=&\, A_{11},\quad B_{s}=B_{1},\nonumber\\ Q_{s}=&\, Q_{11},\quad S_{s}=S_{1}-Q_{12}B_{2},\quad R_{s}=B_{2}^{\ast}Q_{22}B_{2} - B_{2}^{\ast}S_{2}-S_{2}^{\ast}B_{2}+R. \end{align} (18) Proposition 3.4 Consider the Popov function $$ \Phi_{F}(z) := \left[\!\!\begin{array}{c} \left(zE_{F}-A_{F}\right)^{-1}B_{F} \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} Q_{F} & S_{F} \\ S_{F}^{\ast} & R_{F} \end{array}\!\!\right] \left[\!\!\begin{array}{c} (zE_{F}-A_{F})^{-1}B_{F} \\ I_{m} \end{array}\!\!\right] \in \mathbb{K}[z]^{m\times m} $$ of the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17). (a) The Popov functions ΦF(z) and Φ(z) are related via $$ \Phi_{F}(z)=\Theta_{F}^{\sim}(z)\Phi(z)\Theta_{F}(z), $$ where $$\Theta _{F}(z)=I_{m}+FT(zE_{F}-A_{F})^{-1}B_{F}\in \mathbb{K}[z]^{m\times m}$$ is invertible. (b) Further, assume that $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$ is given in feedback equivalence form as in (3) and partitioned accordingly. Then, it holds that ΦF(z) = Φs(z), where Φs(z) is the Popov function corresponding to the EDE part $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K})$$ as in (18). Proof. See the studies by Reis et al. (2015), Voigt (2015) and Bankmann (2016, Proposition 3.8). We now turn to a reduction of the problem for systems $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ in feedback equivalence form as in (17) to the corresponding EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18). Lemma 3.5 Assume that $$(E_F, A_F, B_F, Q_F, S_F, R_F)\in \Sigma^w_{m,n}(\mathbb{K}) $$ as in (17) is given in feedback equivalence form as in (3) and partitioned accordingly. Further, consider the corresponding EDE part $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18) and partition the Hermitian matrix $$ P_{F}= \left[\!\!\begin{array}{ccc} P_{11} & P_{12} & P_{13}\\ P_{12}^{\ast} & P_{22} & P_{23}\\ P_{13}^{\ast} & P_{23}^{\ast} & P_{33} \end{array}\!\!\right] \in\mathbb{K}^{n \times n} $$ accordingly. Then, $$P_{11}\in \mathbb{K}^{n_{1}\times n_{1}}$$ is a solution of the KYP inequality (16) corresponding to the EDE part $$(I_{n_1}, A_s, B_s, Q_s, S_s, R_s)$$ if and only if PF is a solution of the KYP inequality (16) corresponding to $$ (E_F, A_F, B_F, Q_F, S_F, R_F)$$. Proof. We have \begin{align} &\left[\!\!\begin{array}{cc} A_{F}^{\ast}P_{F}A_{F} - E_{F}^{\ast}P_{F}E_{F} +Q_{F} & A_{F}^{\ast}P_{F}B_{F} +S_{F} \\ B_{F}^{\ast}P_{F}A_{F} +S_{F}^{\ast} & B_{F}^{\ast}P_{F}B_{F} +R_{F} \end{array}\!\!\right]\nonumber\\ &\quad=\left[ \begin{array}{@{}ccc|c@{}} A_{11}^{\ast}P_{11}A_{11}-P_{11} & A_{11}^{\ast}P_{12} & M_{13} & A_{11}^{\ast}P_{11}B_{1} + A_{11}^{\ast}P_{12}B_{2} \\ P_{12}^{\ast}A_{11} & P_{22} & M_{23} & P_{12}^{\ast}B_{1} + P_{22}B_{2} \\ M_{13}^{\ast} & M_{23}^{\ast} & M_{33} & M_{34} \\\hline B_{1}^{\ast}P_{11}A_{11} + B_{2}^{\ast}P_{12}^{\ast}A_{11} & B_{1}^{\ast}P_{12} +B_{2}^{\ast}P_{22} & M_{34}^{\ast} & M_{44} \end{array}\right] \nonumber\\ &\qquad\quad+ \left[ \begin{array}{@{}ccc|c@{}} Q_{11} & Q_{12} & Q_{13} & S_{1} \\ Q_{12}^{\ast} & Q_{22} & Q_{23} & S_{2} \\ Q_{13}^{\ast} & Q_{23}^{\ast} & Q_{33} & S_{3} \\ \hline S_{1}^{\ast} & S_{2}^{\ast} & S_{3}^{\ast} & R \end{array}\right] \end{align} (19) for some $$M_{13}\in \mathbb{K}^{n_{1}\times n_{3}}$$, $$M_{23}\in \mathbb{K}^{n_{2}\times n_{3}}$$, $$M_{33}\in \mathbb{K}^{n_{3}\times n_{3}}$$, $$M_{34}\in \mathbb{K}^{n_{3}\times m}$$ and $$ M_{44}=B_{1}^{\ast}P_{11}B_{1}+B_{1}^{\ast}P_{12}B_{2} +B_{2}^{\ast}P_{22}B_{2} + B_{2}^{\ast}P_{12}^{\ast}B_{1}\in\mathbb{K}^{m\times m}. $$ Let $$({x^{*},u^{*}})^{*}\in \mathcal{V}_{(E_F,A_F,B_F)} $$. Thus, by (6) there exists an $$x _{1}\in \mathbb{K}^{n_{1}}$$ such that $$ x=\left(\begin{array}{@{}c@{}} x_{1}\\ -B_{2}u\\ 0_{n_{3}\times1} \end{array}\right). $$ Then, we obtain \begin{multline*} \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{F}^{\ast}P_{F}A_{F} - E_{F}^{\ast}P_{F}E_{F} + Q_{F} & A_{F}^{\ast}P_{F}B_{F} + S_{F} \\ B_{F}^{\ast}P_{F}A_{F} + S_{F}^{\ast} & B_{F}^{\ast}P_{F}B_{F} + R_{F} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)\\ \qquad= \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{s}^{\ast}P_{11}A_{s}-P_{11}+Q_{s} & A_{s}^{\ast}P_{11}B_{s} + S_{s} \\ B_{s}^{\ast}P_{11}A_{s} + S_{s}^{\ast} & B_{s}^{\ast}P_{11}B_{s} + R_{s} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right). \end{multline*} Thus, $$ \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{s}^{\ast}P_{11}A_{s}-P_{11}+Q_{s} & A_{s}^{\ast}P_{11}B_{s} + S_{s} \\ B_{s}^{\ast}P_{11}A_{s} + S_{s}^{\ast} & B_{s}^{\ast}P_{11}B_{s} + R_{s} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x_{1}\\ u \end{array}\!\!\right) \ge 0 $$ for all $$({x_{1},u})\in \mathbb{K}^{n_{1}+m}$$ if and only if $$ \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)^{\ast} \left[\!\!\begin{array}{cc} A_{F}^{\ast}P_{F}A_{F} - E_{F}^{\ast}P_{F}E_{F} + Q_{F} & A_{F}^{\ast}P_{F}B_{F} + S_{F} \\ B_{F}^{\ast}P_{F}A_{F} + S_{F}^{\ast} & B_{F}^{\ast}P_{F}B_{F} + R_{F} \end{array}\!\!\right] \left(\!\!\begin{array}{c} x\\ u \end{array}\!\!\right)\ge0 $$ for all $$({x^{*},u^{*}})^{*}\in \mathcal{V}_{(E_F,A_F,B_F)}$$. Hence, P11 is a solution of the KYP inequality (16) corresponding to the EDE part if and only if PF solves (16) corresponding to $$(E_{F},A_{F},B_{F})$$. We are now ready to state the generalization of the KYP lemma for IDEs. Theorem 3.6 (KYP lemma for IDEs) Let $$(E, A, B, Q, S, R)\in \Sigma^w_{m,n}(\mathbb{K}) $$ be given with corresponding Popov function $$\Phi (z)\in \mathbb{K}[z]^{m\times m}$$. (a) If there exists some $$P\in \mathbb{K}^{n \times n}$$ that is a solution of (16), then $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right ) \succeq 0$$ for all $$\omega \in \mathbb{R}$$ with $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. (b) If on the other hand (E, A, B) is R-controllable and $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right ) \succeq 0$$ for all $$\omega \in \mathbb{R}$$ with $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$, then there exists a solution $$P\in \mathbb{K}^{n \times n}$$ of (16). Proof. We first show assertion (a). Assume that $$P\in \mathbb{K}^{n \times n}$$ fulfils the KYP inequality (16), i.e. $$\mathcal{M}(P)\succeq_{\,\mathcal{V}_{(E,A,B)} }\!0$$. Further, let $$\omega \in \mathbb{R}$$ be such that $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. Then, by Lemma 3.2, together with Proposition 2.11(b), statement (a) follows due to \begin{align*} \Phi\left(\mathrm{e}^{\mathrm{i}\omega}\right)=& \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\sim} \left[\!\!\begin{array}{cc} Q & S \\ S^{\ast} & R \end{array}\!\!\right] \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right]\\ =& \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right]^{\ast} \mathcal{M}(P) \left[\!\!\begin{array}{c} \left(\mathrm{e}^{\mathrm{i}\omega} E-A\right)^{-1}B \\ I_{m} \end{array}\!\!\right] \succeq 0. \end{align*} For part (b) assume that $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all $$\omega \in \mathbb{R}$$ with $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E-A\right )\neq 0$$. For the system in feedback equivalence form $$(E_{F},A_{F},B_{F},Q_{F},S_{F},R_{F}) \in \Sigma^w_{m,n}(\mathbb{K})$$ and corresponding Popov function $$\Phi _{F}(z)\in \mathbb{K}[z]^{m\times m}$$ we obtain from Proposition 3.4(b) that $$\Phi _{F}\left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ for all $$\omega \in \mathbb{R}$$ also fulfilling $$\det \left (\mathrm{e}^{\mathrm{i}\omega} E_{F}-A_{F}\right )\neq 0$$. In particular, by Proposition 2.8 for such ω we have $${\det\!\left(\mathrm{e}^{\mathrm{i}\omega} I_{n_{1}}-A_{11}\right )\neq 0}$$. Furthermore, by Proposition 2.13(b) the associated EDE system $$(I_{n_{1}}, A_{11}, B_{1})$$ is controllable. This means we are in the situation of this theorem for EDEs (Rantzer, 1996) for the EDE system $$ (I_{n_1}, A_s, B_s, Q_s, S_s, R_s)\in \Sigma^w_{m,n_1}(\mathbb{K}) $$ as in (18). Thus, applying Lemma 3.5 gives a solution PF of the KYP inequality (16) corresponding to the system $$(E_F, A_F, B_F, Q_F, S_F, R_F)$$. Then, using Lemma 3.3 completes the proof. Example 3.7 (Example 2.7 revisited) Consider the system $$(E, A, B)$$ as in Example 2.7. From its feedback equivalence form as in (5) we obtain with (6) that $$ V_{F}= \left[\!\!\begin{array}{ccc} 1 & 0 & 0\\ 0 & 0 & -1\\ 0 & 0 & 1 \end{array}\!\!\right] $$ spans the system space $$\mathcal{V}_{(E_F,A_F,B_F)}$$ and thus $$ \mathcal{V}_{(E,A,B)} = \mathcal T_{F} \mathcal{V}_{(E_F,A_F,B_F)}= \operatorname{im} \left[\!\!\begin{array}{ccc} 1 & 0 & -1\\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{array}\!\!\right]\quad \textrm{with}\quad \mathcal T_{F}= \left[\!\!\begin{array}{ccc} 1 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 1 \end{array}\!\!\right]. $$ From $$ \left[\!\!\begin{array}{cc} Q & S\\ S^{\ast} & R \end{array}\!\!\right]=I_{3} $$ we obtain as modified weights $$ \left[\!\!\begin{array}{cc} Q_{F} & S_{F}\\ S_{F}^{\ast}& R_{F} \end{array}\!\!\right]= \left[\!\!\begin{array}{cc} T^{\ast}\left(Q + F^{\ast}S^{\ast} + SF + F^{\ast}RF\right)T & T^{\ast}\left(S+F^{\ast}R\right)\\ \left(S^{\ast}+RF\right)T & R \end{array}\!\!\right] = \left[\!\!\begin{array}{ccc} 2 & 1 & 0\\ 1 & 1 & 0\\ 0 & 0 & 1 \end{array}\!\!\right]. $$ Moreover, the associated EDE part as in (18) is given by \begin{align} A_{s}=1,\quad B_{s}=-1,\quad Q_{s}= 2,\quad S_{s}=-1,\quad R_{s}=2. \end{align} (20) Thus, P11 solves the KYP inequality $$ \left[\!\!\begin{array}{cc} 2 & -P_{11}-1 \\ -P_{11}-1 & P_{11}+2 \end{array}\!\!\right]\succeq 0 $$ if and only if $$ -\sqrt{3}\le P_{11}\le \sqrt{3}. $$ Therefore, choosing P11 = −1, we have that $$ P=W^{\ast}P_{F}W= \left[\!\!\begin{array}{cc} 1 & -1\\ 1 & 0 \end{array}\!\!\right] \left[\!\!\begin{array}{cc} -1 & 0\\\ 0 & 0 \end{array}\!\!\right] \left[\!\!\begin{array}{cc} 1 & 1\\ -1 & 0 \end{array}\!\!\right] = \left[\!\!\begin{array}{cc} -1 & -1\\ -1 & -1 \end{array}\!\!\right] $$ solves the KYP inequality (16). In particular, by Theorem 3.6 we obtain that for the Popov functions $$\Phi _{F}(z)\in \mathbb{K}(z)$$ and $$\Phi (z)\in \mathbb{K}(z)$$ we have $$\Phi _{F}\left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$ and $$\Phi \left (\mathrm{e}^{\mathrm{i}\omega} \right )\succeq 0$$. Remark 3.8 The result of Theorem 3.6 is analogous to the continuous-time result in the study by Reis et al. (2015). To see this, replace positivity of the Popov function on the unit circle by positivity on the imaginary axis in (a) and replace $$\mathcal M(P)$$ by its continuous-time analogue. However, in the study by Reis et al. (2015) the assumption of R-controllability was alternatively replaced by the condition that the Popov function has full rank and $$(E, A, B)$$ is sign-controllable. To adapt this to the discrete-time setting we would need a discrete-time analogue of Clements et al. (1997, Theorem 6.1), which provides the characterizations via sign-controllability in the ODE case. 4. Structure of palindromic matrix pencils In this section we are concerned with palindromic matrix pencils $$z\mathcal{A}^{\ast } - \mathcal{A} \in \mathbb{K}[z]^{n\times n}$$. For the investigation of these palindromic matrix pencils we first introduce so-called quasi-Hermitian matrices. Then, we show characterizations of the inertia of palindromic matrix pencils similar to what was done in the studies by Clements & Glover (1989), Clements et al. (1997), Reis (2011) and Voigt (2015), in the case of so-called even matrix pencils. The concept of quasi-Hermitian matrices is an extension to the notion of Hermitian and skew-Hermitian matrices. These are matrices $$\mathcal{A}\in \mathbb{K}^{n \times n}$$ with the property $$\mathcal{A}=\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast }$$ for some ω ∈ [0, 2π). They have the special property that every eigenvalue lies on the line with angle ω/2 through the origin. We can extend the notion of inertia for Hermitian matrices to quasi-Hermitian matrices. Definition 4.1 Let $$\mathcal{A}=\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast }$$ be quasi-Hermitian with ω ∈ [0, 2π). Then, the inertia of $$\mathcal{A}$$ along $$\frac{\omega }{2}$$ is $$ \operatorname{In}(\mathcal{A}) := \operatorname{In}_{\frac{\omega}{2}}(\mathcal{A}):=(n_{+},\,n_{0},\,n_{-}), $$ where n+, n0 and n− denote the number of eigenvalues $$\lambda =r\mathrm{e}^{\mathrm{i}\frac{\omega }{2}}$$ where r is positive, zero or negative, respectively. We omit the subscript $$\frac{\omega }{2}$$ in $$ \operatorname{In}_{\frac{\omega }{2}}(\mathcal{A})$$ if the angle is clear from the context. Similar to the Hermitian case, also in the quasi-Hermitian case we have a canonical form under congruence transformations. Theorem 4.2 The inertia of a quasi-Hermitian matrix is invariant under congruence transformations. On the other hand, if $$\mathcal{A},\,\mathcal{B}\in \mathbb{K}^{n \times n}$$ are two quasi-Hermitian matrices having the same inertia with respect to the same ω ∈ [0, 2π), then there exists some invertible $$U\in \mathbb{K}^{n \times n}$$ such that $$ U^{\ast}\mathcal{A} U=\mathcal{B}, $$ i.e. $$\mathcal{A}$$ and $$\mathcal{B}$$ are congruent. Proof. See the article by Ikramov (2001). We are now interested in a structure-preserving canonical form revealing the eigenstructure of a palindromic matrix pencil. Theorem 4.3 (Palindromic Kronecker canonical form) (Schröder, 2008) Let $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$ be a palindromic matrix pencil. Then, there exists some invertible $$U\in \mathbb{C}^{n\times n}$$ such that \begin{align} U^{\ast}\left(z\mathcal{A}^{\ast}-\mathcal{A}\right)U = \operatorname{diag}\left({D_{1}(z),\ldots,D_{l}(z)} \right) \end{align} (21) for some $$l\in \mathbb{N}$$ is in palindromic Kronecker canonical form (PKCF), where each block $$D_{j}(z)\in \mathbb{C}[z]^{k_{j}\times k_{j}}$$, $$k_{j}\in \mathbb{N}$$, is of one of the forms shown in Table 3 and The PKCF is unique up to permutations of the blocks and the quantities εj ∈{−1, 1} are called sign-characteristics. A closely related version of the above theorem was developed in the article by Horn & Sergeichuk (2006). Remark 4.4 We have multiplied the sign-characteristics of the blocks of type P4 occurring in the thesis by Schröder (2008) with −1 in order to simplify some of the upcoming results. This is justified by the fact that if $$\tilde{D}_{j}(z)$$ with sign-characteristic $$\tilde{\varepsilon }_{j}$$ corresponds to a block of type P4 introduced in the thesis by Schröder (2008), then $$D_{j}(z)=-U^{\ast }\tilde{D}_{j}(z)U$$ with $$ U=\mathrm{i} \left[\begin{array}{@{}cc@{}} I_{k_{j}/2} & \\ & -I_{k_{j}/2} \end{array}\right]\in\mathbb{C}^{k_{j}\times k_{j}} $$ is a block of type P4 with sign-characteristic $$\varepsilon _{j}=-\tilde{\varepsilon }_{j}$$ according to Theorem 4.3. Remark 4.5 By analysing the eigenstructure of the blocks in the form (21) we obtain (a) Blocks of type P1 correspond to eigenvalues λ and $${1}/{\overline{\lambda }}$$ with $$|\lambda |\neq 1$$, i.e. these eigenvalues occur in pairs $$\left \{\lambda ,{1}/{\overline{\lambda }}\right \}$$. In particular, this holds for the pairing $$\{0,\infty \}$$. (b) Blocks of type P2, P3 and P4 correspond to eigenvalues λ with $$\vert\lambda\vert=1 $$. (c) Blocks of type P5 correspond to rank deficiency of the pencil, i.e. they correspond to singular blocks. Consider the palindromic matrix pencil $$\mathcal{P}(z)=z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$. By inserting $$\mathrm{e}^{\mathrm{i}\omega} $$ for the polynomial variable z we obtain $$ \mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right)=\mathrm{e}^{\mathrm{i}\omega} \mathcal{A}^{\ast}-\mathcal{A} = \mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\left(\mathrm{i} \mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\mathcal{A}-\mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\mathcal{A}^{\ast}\right) $$ and hence $$ \mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right)^{\ast}= -\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\left(\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\mathcal{A}-\mathrm{i}\mathrm{e}^{\mathrm{i}\frac{\omega}{2}}\mathcal{A}^{\ast}\right)= \left(-\mathrm{i}\mathrm{e}^{-\mathrm{i}\frac{\omega}{2}}\right)^{2}\mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right)=-\mathrm{e}^{-\mathrm{i}\omega} \mathcal{P}\left(\mathrm{e}^{\mathrm{i}\omega}\right). $$ Thus, $$\mathcal{P}\left (\mathrm{e}^{\mathrm{i}\omega} \right )$$ is quasi-Hermitian and has well-defined inertia. Investigating the block structure of the PKCF leads to the following result: Lemma 4.6 Assume that $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$ is in PKCF, i.e. it holds that $$z\mathcal{A}^{\ast }-\mathcal{A} = \operatorname{diag}\left ({D_{1}(z),\ldots ,D_{l}(z)} \right )$$ for some $$l\in \mathbb{N}$$. Then, the inertia pattern of each block $$D_{j}(z)\in \mathbb{C}[z]^{k_{j}\times k_{j}}$$, $$k_{j}\in \mathbb{N}$$, is given as in Table 3. Proof. See the thesis by Bankmann (2016, Lemma 4.11). Remark 4.7 The results from Lemma 4.6 can be used to determine the block structure of a pencil $$z\mathcal{A}^{\ast }-\mathcal{A}\in \mathbb{K}[z]^{n\times n}$$ in the form (21), given the inertia patterns for ω ∈ [0, 2π). Note that blocks of type P1 have a very simple inertia pattern and thus from a general pattern $$ \operatorname{In}\left(\mathrm{e}^{\mathrm{i}\omega}\mathcal{A}^{\ast}-\mathcal{A}\right) = \left(k_{1},k_{2},k_{3}\right) $$ —except for the case where k2 = 0, i.e. all blocks are of type P1—we cannot tell whether or how many blocks of type P1 are present in the PKCF. Table 3. Blocks D(z) occurring in palindromic Kronecker canonical form and their inertia $$ \operatorname{In}\left ( D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ for ω ∈ [0, 2π), where the addition of two inertia tuples has to be understood component-wise Type Dimension D(z) Parameters $$ \operatorname{In}\left (D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ P1 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda )\\\hline zJ_{k}(\overline{\lambda} )-F_{k} &\\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon \!\in\! \{-1,1\},\\ \lambda \!=\!r\mathrm{e}^{\mathrm{i}\theta },\,{|r|<1,}\\{\theta \!\in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c} \left (k,0,k\right )\\[23pt]\end{array}$$ P2 2k + 1 $$\varepsilon \left [ \begin{array}{c|c|c} & & zF_{k}-J_{k}(\lambda )\\\hline & z\mathrm{e}^{-\mathrm{i}\frac{\theta }{2}}- \mathrm{e}^{\mathrm{i}\frac{\theta }{2}} & -{\left (e^{ k}_{1}\right )}^{T}\\\hline zJ_{k}(\overline \lambda )-F_{k} & ze^{ k}_{1} \\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon\! \in\! \{-\!1,1\},\\ \lambda \!=\!\mathrm{e}^{\mathrm{i}\theta }\!,{\theta\! \in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c}\left (k,0,k\right )\!+\! \operatorname{In}\!\left (\varepsilon (\omega \!-\!\theta )\right )\\[11pt]\end{array}$$ P3 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda ) \\\hline zJ_{k}(\overline \lambda )-F_{k} & (z-1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right] $$ $$\begin{array}{l}{k\in \mathbb{N}_{0},}\,\varepsilon \in \{-1,1\},\\ \lambda =\mathrm{e}^{\mathrm{i}\theta }\!,{\theta \in [0,2\pi )}\end{array}$$ $$\begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq \theta \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = \theta \end{cases} $$ P4 2k $$\varepsilon \left [ \begin{array}{c|c} & zF_{k}-J_{k}(1)\\\hline zJ_{k}(1)-F_{k} & \textrm{i}(z+1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right ]$$ $$\ k\in \mathbb{N}_{0},\,\!\varepsilon \!\in\! \{-1,1\}$$ $$ \begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq 0 \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = 0 \end{cases} $$ P5 2k + 1 $$\left [ \begin{array}{c|c} & zS_{R} - {S_{L}^{T}} \\\hline zS_{L} - {S_{R}^{T}} & \\ \end{array}\right ] $$ $$\ k\in \mathbb{N}_{0}$$ $$\ \left (k,1,k\right )$$ Type Dimension D(z) Parameters $$ \operatorname{In}\left (D\left (\mathrm{e}^{\mathrm{i}\omega} \right )\right )$$ P1 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda )\\\hline zJ_{k}(\overline{\lambda} )-F_{k} &\\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon \!\in\! \{-1,1\},\\ \lambda \!=\!r\mathrm{e}^{\mathrm{i}\theta },\,{|r|<1,}\\{\theta \!\in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c} \left (k,0,k\right )\\[23pt]\end{array}$$ P2 2k + 1 $$\varepsilon \left [ \begin{array}{c|c|c} & & zF_{k}-J_{k}(\lambda )\\\hline & z\mathrm{e}^{-\mathrm{i}\frac{\theta }{2}}- \mathrm{e}^{\mathrm{i}\frac{\theta }{2}} & -{\left (e^{ k}_{1}\right )}^{T}\\\hline zJ_{k}(\overline \lambda )-F_{k} & ze^{ k}_{1} \\ \end{array}\right ] $$ $$\begin{array}{l}{k\!\in\! \mathbb{N}_{0},}\,\varepsilon\! \in\! \{-\!1,1\},\\ \lambda \!=\!\mathrm{e}^{\mathrm{i}\theta }\!,{\theta\! \in\! [0,2\pi )}\end{array}$$ $$\begin{array}{c}\left (k,0,k\right )\!+\! \operatorname{In}\!\left (\varepsilon (\omega \!-\!\theta )\right )\\[11pt]\end{array}$$ P3 2k $$\varepsilon \left [\begin{array}{c|c} & zF_{k}-J_{k}(\lambda ) \\\hline zJ_{k}(\overline \lambda )-F_{k} & (z-1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right] $$ $$\begin{array}{l}{k\in \mathbb{N}_{0},}\,\varepsilon \in \{-1,1\},\\ \lambda =\mathrm{e}^{\mathrm{i}\theta }\!,{\theta \in [0,2\pi )}\end{array}$$ $$\begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq \theta \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = \theta \end{cases} $$ P4 2k $$\varepsilon \left [ \begin{array}{c|c} & zF_{k}-J_{k}(1)\\\hline zJ_{k}(1)-F_{k} & \textrm{i}(z+1){{e^{k}_{1}}}{\left (e^{ k}_{1}\right )}^{T} \end{array}\right ]$$ $$\ k\in \mathbb{N}_{0},\,\!\varepsilon \!\in\! \{-1,1\}$$ $$ \begin{cases} \left (k,0,k\right ), \qquad\qquad\qquad\quad\ \ \omega \neq 0 \\ \left (k-1,1,k-1\right ) + \operatorname{In}(\varepsilon ),\quad \omega = 0 \end{cases} $$ P5 2k + 1 $$\left [ \begin{array}{c|c} & zS_{R} - {S_{L}^{T}} \\\hline zS_{L} - {S_{R}^{T}} & \\ \end{array}\right ] $$ $$\ k\in \mathbb{N}_{0}$$ $$\ \left (k,1,k\right )$$ View Large Table 3. Blocks D(z) occurring in palindromic Kronecker canonical form and their iner