A new method for proving the separation principle for the infinite-dimensional LQG regulator problem

A new method for proving the separation principle for the infinite-dimensional LQG regulator problem Abstract This paper gives an alternative perspective on the standard linear quadratic Gaussian regulator problem for infinite-dimensional state-space systems. We will show that when considered on an extended Hilbert state-space, the originally stochastic control problem can be phrased as a deterministic one. In this setting we obtain a new method of proof for the well-known separation principle which does not involve probabilistic considerations. 1. Introduction A frequent occurrence in practical distributed parameter systems, with finite-dimensional inputs and outputs, is for them to be controlled by deriving optimal controls (if they exist) for the system subject not only to additive stochastic disturbances, but also to partial observations of the state. This is commonly known as the ‘partially observable optimal control problem’. If the system to be controlled is linear and the cost quadratic, then the minimization called for in the stochastic control problem will be a problem that clearly falls within the scope of the standard linear quadratic Gaussian (LQG) regulator problem and, if solved successfully, will lead to the construction of an optimal state estimate feedback control law relative to the quadratic cost functional involved. At the heart of the solution to the LQG regulator problem lies the so-called separation principle (originally stated and proven by Wonham, 1968b for finite-dimensional systems) which—in both infinite and finite dimensions—is normally investigated using abstract probability theory. The results are, of course, very well known, and for infinite-dimensional systems a few approaches are available; see the texts by Curtain & Pritchard (1978), Balakrishnan (1981) and the relevant references cited there. In the present paper we will be calling for the proof of the separation principle with the motivation of avoiding a probabilistic treatment. Some work has been done along similar lines; yet, an early work which might be considered most closely related to ours is probably that of Ahmed & Li (1991) (see also the book by Ahmed, 1998). Although they use a similar technique in the finite-dimensional case to reformulate and solve the individual problems of state estimation and feedback control inherent in the LQG regulator problem, they do not explicitly consider the joint problem of state estimate feedback control. We shall see here, however, that the technical details required for this will only be slightly more complex, even for infinite-dimensional systems, when the controls are assumed from the beginning to be of the standard state estimate feedback type. In our developments in this paper, in Section 2 to be precise, the LQG regulator problem is posed not on the usual Hilbert state-space, but rather on an extended Hilbert space. This allows us to convert the controlled stochastic system—in fact, a suitable ‘augmentation’ of it—into a deterministic one and reformulate the cost functional so that the dependence of the new cost functional will be on the feedback and estimator gain operators (definitions in Section 2) rather than only on the control inputs. In so doing, the new minimization problem, as we shall see, becomes considerably simpler than the original one in that it will be amenable to treatment within the framework of the variational calculus of operators. So by the end of Section 2 we will be well prepared to examine the question of necessary conditions. In Section 3 our intention, broadly put, will be to derive a solution to our minimization problem. We will not be concerned with what is mathematically a not-so-trivial problem, namely, the problem of existence of the optimal pair of feedback and estimator gain operators in spaces of operator functions. Rather, we will derive a set of necessary conditions and thereby provide a different approach to the proof of the separation principle in infinite dimensions, which—we already touched upon this—does not rely on a probabilistic treatment. In point of fact, our main achievement there will be to establish the separation principle without employing the theory of Wiener processes and stochastic integration and hence to offer an alternative route to generalizing the LQG regulator problem to an infinite-dimensional setting. Our method of proof is new, and we believe that not only does it complement the direct approach as considered, for example, by Balakrishnan in his book, but is interesting for another reason: it has appealing mathematical simplicity because it relies almost exclusively on the variational arguments. The solution for the feedback and estimator gain operators entails two sets of four non-standard, coupled differential operator equations each, which can be shown by simple arguments to reduce to the well-known Riccati equations that arise in the solution to the LQG regulator problem. The final result in Section 3 then usefully indicates how an existence theorem for the control problem (which, as we have noted earlier, we do not present here) would imply the uniqueness of the optimal pair of feedback and estimator gain operators thus found. As a final note in closing, the approach presented in this paper differs, somewhat conceptually, from that of most of the literature in that the structure of the state estimator is fixed beforehand. The state estimator plays a central role in forming the augmented system, and this means in effect that the equations describing the state estimator become part of the control problem formulation, strictly speaking, and not the solution. 2. An equivalent deterministic form of the LQG regulator problem We will deal in this paper with systems of rather general type described by (strongly continuous) $$C_{0}$$-semigroups and evolution families, and we begin by briefly formulating the standard LQG regulator problem as it applies to linear, time-invariant, stochastic dynamical systems in an infinite-dimensional Hilbert space setting. We will use in this section (and in fact throughout the whole paper) some of the notation and results of the book by Curtain & Zwart (1995) to which we refer the reader for more details. Consider any time interval $$\left [t_{0},t_{1}\right ]\subset \mathbb{R}_{+}$$, with $$0\leqslant t_{0} < t_{1}$$, wherein the controlled system is to evolve. We start with consideration of the abstract system   \begin{align} \dot{x}\left(t\right)&=Ax\left(t\right)+Bu\left(t\right)+Dv\left(t\right),\quad t_{0} < t < t_{1},\nonumber\\[-3pt] \quad x\left(t_{0}\right)&=x_{0}, \end{align} (2.1)on some real separable Hilbert state-space X, accompanied by the output equation   \begin{align} y\left(t\right)=Cx\left(t\right)+w\left(t\right),\quad t_{0}\leqslant t\leqslant t_{1}. \end{align} (2.2)The system operator $$A:\boldsymbol{D}\left (A\right )\left (\subset X\right )\rightarrow X$$, which is (as is usually the case in physical dynamical systems) maximal dissipative, thus closed and densely defined, and hence is the infinitesimal generator of a $$C_{0}$$-semigroup $$T\left (\;\!\cdot \;\!\right ):\left [t_{0},t_{1}\right ]\rightarrow \mathcal{L}\left (X\right )$$ of contractions. For each fixed $$x_{0}\in X$$ one then has $$T\left (\;\!\cdot \;\!\right )x_{0}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];X\right )$$, i.e. the map $$t\mapsto T\left (t\right )$$ is continuous in the strong topology of $$\mathcal{L}\left (X\right )$$. We assume the mappings B, C and D are all finite-dimensional, in the sense that they have finite rank and are bounded, $$B,D\in \mathcal{L}\left (\mathbb{R}^{m},X\right )$$ and $$C\in \mathcal{L}\left (X,\mathbb{R}^{p}\right )$$. As usual, we will allow for discontinuous control inputs and thus assume $$u\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$. To ‘randomize’ the system in some sense, and in accordance with the standard LQG regulator problem statement, let it be assumed that the initial state $$x_{0}$$ is a Gaussian random variable, having zero mean, and covariance operator $$S_{0}\in \mathcal{L}\left (X\right )$$, which we will not assume to be necessarily nuclear. We will take into account the functions $$v\left (\;\!\cdot \;\!\right )$$ and $$w\left (\;\!\cdot \;\!\right )$$ as random entities also, which correspond to what is called ‘Gaussian white noise’ in the literature, but without having to resort to the theory of stochastic integration for Wiener processes. A rigorous justification for doing so rests with Balakrishnan’s formalism of white noise theory (in his book mentioned in the introduction) so full details are not necessary here. Suffice it to say that $$v\left (\;\!\cdot \;\!\right ),w\left (\;\!\cdot \;\!\right )$$ have sample paths which are norm square-integrable on $$\left [t_{0},t_{1}\right ]$$, i.e. they are, respectively, elements of $$\boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$ and $$\boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{p}\right )$$ and such that for any $$f\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$ and $$g\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{p}\right )$$ the quantities (where $$\left\langle \;\!\cdot \;\!,\;\!\cdot \;\!\right\rangle $$ denotes the usual Euclidean inner product)   $$ \int_{t_{0}}^{t_{1}} \left\langle v\left(t\right),f\left(t\right)\right\rangle \,\mathrm{d}t\quad\textrm{and}\quad\int_{t_{0}}^{t_{1}} \left\langle w\left(t\right),g\left(t\right)\right\rangle \,\mathrm{d}t $$are Gaussian with zero mean and positive symmetric covariance matrices. We take the matrices $$V\in \mathcal{L}\left (\mathbb{R}^{m}\right )$$ and $$W\in \mathcal{L}\left (\mathbb{R}^{p}\right )$$ as the covariances of $$v\left (\;\!\cdot \;\!\right )$$ and $$w\left (\;\!\cdot \;\!\right )$$, respectively, but to prove the direct analogue of the separation principle we will not, in general, consider the case where W is only positive. Rather, we shall require W to be bounded below (and hence boundedly invertible), so $$W^{-1}\in \mathcal{L}\left (\mathbb{R}^{p}\right )$$. This corresponds to the case of entirely non-perfect observations of the state. Further, and this is our final assumption, $$v\left (\;\!\cdot \;\!\right )$$, $$w\left (\;\!\cdot \;\!\right )$$ and $$x_{0}$$ are assumed mutually independent. The initial state $$x_{0}$$ can be any element of X, and because $$u\left (\;\!\cdot \;\!\right ),v\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$, we have only the weak or mild version of (2.1), which is written in terms of $$T\left (\;\!\cdot \;\!\right )$$,   \begin{align} x\left(t\right)=T\left(t-t_{0}\right)x_{0}+\int^{t}_{t_{0}}T\left(t-{s}\right)Bu\left({s}\right)\,\mathrm{d}{s}+\int^{t}_{t_{0}}T\left(t-{s}\right)Dv\left({s}\right)\,\mathrm{d}{s},\quad t_{0}\leqslant t\leqslant t_{1}. \end{align} (2.3)We therefore define the integral equation (2.3) to be the generalized solution of (2.1), which exists and is unique, and $$x\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];X\right )$$. Moreover, under our assumptions on B, D and $$x_{0}$$, the state $$x\left (t\right )$$ is a Gaussian random variable with zero mean for each fixed value of t. In the paper we shall almost always work with the differential notation (2.1), bearing in mind that what we really mean is the integral version (2.3). Suppose that the controls $$u\left (\;\!\cdot \;\!\right )$$ are ‘admissible’ if $$u\left (t\right )$$ is, in a sense that will be made more precise shortly, based on the output as defined by (2.2) up to time t for $$t_{0}\leqslant t\leqslant t_{1}$$. Given then the operator $$Q\in \mathcal{L}\left (X\right )$$, which is positive symmetric and nuclear, and some positive symmetric matrix $$R\in \mathcal{L}\left (\mathbb{R}^{m}\right )$$, bounded below so that $$R^{-1}\in \mathcal{L}\left (\mathbb{R}^{m}\right )$$, we associate with (2.3) the so-called cost of $$u\left (\;\!\cdot \;\!\right )$$, defined by   \begin{align} {J}\left(u\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}E\left\langle Qx\left(t_{1}\right),x\left(t_{1}\right)\right\rangle +\frac{1}{2}\int_{t_{0}}^{t_{1}}\left[E\left\langle Qx\left(t\right),x\left(t\right)\right\rangle + E\left\langle Ru\left(t\right),u\left(t\right)\right\rangle \right]\,\mathrm{d}t, \end{align} (2.4)wherein E denotes the expected value. (We have not distinguished in the notation above between either of the inner products on the spaces considered as this is clear from the context.) Let it, then, be required to find the admissible controls $$u\left (\;\!\cdot \;\!\right )$$ which minimize $${J}\left (u\left (\;\!\cdot \;\!\right )\right )$$. This is the well-known LQG regulator problem relative to an abstract semigroup formulation of the underlying system dynamics in Hilbert space. It is now possible to convert the stochastic control problem in a fairly straightforward way into an equivalent deterministic one. This can be done, as we now show, under the assumption that we are to develop a linear state estimate feedback form of the solution to the problem. Suppose to this end that we are given a state estimator, essentially of the Luenberger type, for the system consisting of (2.1) and (2.2):   \begin{align} \dot{\hat{x}}\left(t\right)&=A\hat{x}\left(t\right)+Bu\left(t\right)+L\left(t\right)(y\left(t\right)-C\hat{x}\left(t\right)),\quad t_{0} < t < t_{1},\nonumber\\[-2pt] \hat{x}\left(t_{0}\right)&=0, \end{align} (2.5)where the map $$t\mapsto L\left (t\right )$$ is strongly continuous from $$\left [t_{0},t_{1}\right ]$$ into $$\mathcal{L}\left (\mathbb{R}^{p},X\right )$$. The operator L(t) will be called the ‘estimator gain operator’. Now suppose that to each t is associated another operator $$K\left (t\right )\in \mathcal{L}\left (X,\mathbb{R}^{m}\right )$$, the map $$t\mapsto{K}\left (t\right )$$ being strongly continuous on $$\left [t_{0},t_{1}\right ]$$ also. The work throughout this paper is carried out under the assumption that the control $$u\left (t\right )$$ is always generated via the feedback law   \begin{align} u\left(t\right)=K\left(t\right)\hat{x}\left(t\right),\quad t_{0}\leqslant t\leqslant t_{1}; \end{align} (2.6)the operator K(t) is called the feedback gain operator. Notice that both K(t) and L(t) are unknown operators at this stage. If we now proceed formally, substituting for u(t) from (2.6) into (2.1), we obtain   $$ \dot{x}\left(t\right)=Ax\left(t\right)+BK\left(t\right)\hat{x}\left(t\right)+Dv\left(t\right). $$Similarly, by substituting (2.6) into (2.5) and using (2.2),   $$ \dot{\hat{x}}\left(t\right)=\left(A+BK\left(t\right)\right)\hat{x}\left(t\right)+L\left(t\right)C(x\left(t\right)-\hat{x}\left(t\right))+L\left(t\right)w\left(t\right). $$We then define the estimation error   $$ \tilde{x}\left(t\right):= x\left(t\right)-\hat{x}\left(t\right), $$in terms of which we arrive at   \begin{align} \dot{\tilde{x}}\left(t\right)&=\left(A-L\left(t\right)C\right)\tilde{x}\left(t\right)+Dv\left(t\right)-L\left(t\right)w\left(t\right),\quad t_{0} < t < t_{1},\nonumber\\[-2pt]\quad{\tilde{x}}\left(t_{0}\right)&={\tilde{x}}_{0}, \end{align} (2.7)where $${\tilde{x}}_{0}=x_{0}$$. What we now do is to introduce the product space   $$ Z:= X\oplus X $$as the new state space, namely, the augmented state space. Consider the vectors   $$ {{z}}\left(t\right):=\left(\begin{array}{@{}c@{}} \hat{{x}}\left(t\right) \\ \tilde{{x}}\left(t\right) \end{array}\right)\in Z\quad\textrm{and}\quad{{r}}\left(t\right):=\left(\begin{array}{@{}c@{}} {{v}}\left(t\right) \\{{w}}\left(t\right) \end{array}\right)\in\mathbb{R}^{m+p}, $$and define a ‘new’ system operator $$\mathscr{A}$$ by setting   $$ \mathscr{A}=\left(\begin{array}{@{}cc@{}} A & 0 \\ 0 & A \end{array}\right), $$with domain   \begin{align} \boldsymbol{D}\left(\mathscr{A}\right):=\boldsymbol{D}\left(A\right)\oplus\boldsymbol{D}\left(A\right). \end{align} (2.8)In doing so, as will be seen below, it is implicit that   \begin{align*} \mathscr{B}&=\left(\begin{array}{@{}cc@{}} B & 0 \\ 0 & 0 \end{array}\right),\quad\mathscr{C}=\left(\begin{array}{@{}cc@{}} 0 & 0 \\ 0 & C \end{array}\right),\quad\mathscr{D}=\left(\begin{array}{@{}cc@{}} 0 & 0 \\ D & 0 \end{array}\right),\\ \mathscr{K}\left(t\right)&=\left(\begin{array}{@{}cc@{}} K\left(t\right) & 0 \\ 0 & 0 \end{array}\right),\quad\mathscr{L}\left(t\right)=\left(\begin{array}{@{}cc@{}} 0 & L\left(t\right) \\ 0 & -L\left(t\right) \end{array}\right), \end{align*}and we will frequently make explicit use of these block operator matrices in what is to follow. We let $$\mathscr{B}\left (t\right ):Z\rightarrow Z$$ and $$\mathscr{D}\left (t\right ):\mathbb{R}^{m+p}\rightarrow Z$$ be given by   $$ \mathscr{B}\left(t\right):=\mathscr{B}\mathscr{K}\left(t\right)+\mathscr{L}\left(t\right)\mathscr{C}\quad\textrm{and}\quad\mathscr{D}\left(t\right):=\mathscr{D}+\mathscr{L}\left(t\right), $$where the maps $$t\mapsto \mathscr{B}\left (t\right )$$ and $$t\mapsto \mathscr{D}\left (t\right )$$ are, by virtue of the assumed continuity properties of $${K}\left (\;\!\cdot \;\!\right )$$ and $$L\left (\;\!\cdot \;\!\right )$$, strongly continuous from $$\left [t_{0},t_{1}\right ]$$ into, respectively, $$\mathcal{L}\left (Z\right )$$ and $$\mathcal{L} (\mathbb{R}^{m+p},Z )$$, and so it follows that $$\mathscr{B}\left (\;\!\cdot \;\!\right )$$ and $$\mathscr{D}\left (\;\!\cdot \;\!\right )$$ are uniformly bounded in norm on $$\left [t_{0},t_{1}\right ]$$—i.e. $$\mathscr{B}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (Z\right )\right )$$ and $$\mathscr{D}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty } (\left [t_{0},t_{1}\right ];\mathcal{L} (\mathbb{R}^{m+p},Z ) )$$. The above considerations enable us now to combine (2.5) and (2.7) to write the augmented system as   \begin{align} {\dot{{z}}}\left(t\right)&=\left(\mathscr{A}+\mathscr{B}\left(t\right)\right){{z}}\left(t\right)+\mathscr{D}\left(t\right){{r}}\left(t\right),\quad t_{0} < t < t_{1},\nonumber\\[-2pt] {{z}}\left(t_{0}\right)&=z_{0}, \end{align} (2.9)for the augmented state-space Z, where   $$ z_{0}:=\left(\begin{array}{@{}c@{}} 0 \\ \tilde{x}_{0} \end{array}\right)\in Z. $$Obviously the operator $$\mathscr{A}$$ is the generator of a $$C_{0}$$-semigroup of contractions on Z, and we notice that the operator $$\mathscr{A}+\mathscr{B}\left (t\right )$$, having domain   $$ \boldsymbol{D}\left(\mathscr{A}+\mathscr{B}\left(\;\!\cdot\;\!\right)\right)=\boldsymbol{D}\left(\mathscr{A}\right), $$independent of t, is a bounded linear perturbation of the unperturbed system operator $$\mathscr{A}$$ for each t. Thus, we have that $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ generates a weak evolution family $$\mathscr{U}\left (\;\!\cdot \;\!,\;\!\cdot \;\!\right ):\left \{\left (t,s\right ):t_{0}\leqslant s\leqslant t\leqslant t_{1}\right \}\rightarrow \mathcal{L}\left (Z\right )$$, which possesses the standard properties of a strong evolution family with an important exception (for our purposes at least) that whereas the map $$s\mapsto \mathscr{U}\left (t,{s}\right )$$ is strongly differentiable, the map $$t\mapsto \mathscr{U}\left (t,{s}\right )$$ is not, even though $$\mathscr{B}\left (\;\!\cdot \;\!\right )z_{0}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$. Since $$z_{0}\in Z$$ and $${{r}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2} (\left [t_{0},t_{1}\right ];\mathbb{R}^{m+p} )$$, we therefore conclude that we are again only able to define a generalized solution of (2.9),   \begin{align} {{z}}\left(t\right)=\mathscr{U}\left(t,t_{0}\right)z_{0}+\int^{t}_{t_{0}}\mathscr{U}\left(t,{s}\right)\mathscr{D}\left({s}\right){{r}}\left({s}\right)\,\mathrm{d}{s},\quad t_{0}\leqslant t\leqslant t_{1}, \end{align} (2.10)where $${{z}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$. Then, as before, since $$z_{0}$$ is a Gaussian random variable with zero mean and $$\mathscr{D}\left (t\right )$$ has finite rank for $$t_{0}\leqslant t\leqslant t_{1}$$, the augmented state $${{z}}\left (t\right )$$ is a Gaussian random variable having zero mean for each fixed value of t. We reiterate, as an aside, regarding the interpretation of (2.10) that for $$z_{0}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ the generalized solution is the differentiable solution, in the strong sense, provided that $$\mathscr{B}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );\mathcal{L}\left (Z\right )\right )$$ and $$\mathscr{D}\left (\;\!\cdot \;\!\right ){{r}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );Z\right )$$, i.e. the maps $$t\mapsto \mathscr{B}\left (t\right )$$ and $$t\mapsto \mathscr{D}\left (t\right ){{r}}\left (t\right )$$ are strongly differentiable on $$\left (t_{0},t_{1}\right )$$ (Phillips, 1953). Then the evolution family generated by $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ is a strong evolution family, so the map $$t\mapsto \mathscr{U}\left (t,s\right )z_{0}$$ is differentiable for each $$z_{0}\in \boldsymbol{D}\left (\mathscr{A}\right )$$. This will prove to be important for our sequel. Now, write   $$ \mathscr{S}_{0}=\left(\begin{array}{@{}cc@{}} 0 & 0 \\ 0 & S_{0} \end{array}\right),\quad \mathscr{N}=\left(\begin{array}{@{}cc@{}} V & 0 \\ 0 & W \end{array}\right), $$and denote by $$\mathscr{S}\left (t\right )$$ the covariance operator associated with $${{z}}\left (t\right )$$. Then by way of some straightforward calculations with covariances (Balakrishnan, 1981, p. 317), using the independence of $$z_{0}$$ and $${{r}}\left (\;\!\cdot \;\!\right )$$, we have, for any $$z_{1}\in Z$$, that   \begin{align} \mathscr{S}\left(t\right)z_{1}=\mathscr{U}\left(t,t_{0}\right)\mathscr{S}_{0}\mathscr{U}\left(t,t_{0}\right)^{\ast} z_{1}+\int^{t}_{t_{0}}\mathscr{U}\left(t,{s}\right)\mathscr{D}\left({s}\right)\mathscr{N}\mathscr{D}\left({s}\right)^{\ast}\mathscr{U}\left(t,{s}\right)^{\ast} z_{1}\,\,\mathrm{d}{s}. \end{align} (2.11)Clearly, since $$\mathscr{S}_{0}$$ and $$\mathscr{N}$$ are both positive symmetric, $$\mathscr{S}\left (t\right )$$ is positive symmetric for each t. Moreover, by the uniform boundedness of the norm of the map $$\left (t,s\right )\mapsto \mathscr{U}\left (t,{s}\right )$$ on finite time intervals, we have $$\mathscr{S}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (Z\right )\right )$$. In fact, $${\mathscr{S}}\left (\;\!\cdot \;\!\right )z_{1}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$, and we note in particular that for $$z_{1},z_{2}\in \boldsymbol{D}\left (\mathscr{A}^{\ast }\right )$$ the function $$t\mapsto \left\langle{\mathscr{S}}\left (t\right )z_{1},z_{2}\right\rangle $$—defined as in (2.11)—can be differentiated to satisfy the system (wherein $$\left\langle \;\!\cdot \;\!,\;\!\cdot \;\!\right\rangle $$ now denotes the inner product on Z)   \begin{align} \left\langle \dot{\mathscr{S}}\left(t\right)z_{1},z_{2}\right\rangle &=\left\langle{\mathscr{S}}\left(t\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{2}\right\rangle +\left\langle \left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{1},{\mathscr{S}}\left(t\right)z_{2}\right\rangle\nonumber \\ &\quad+\left\langle \mathscr{N}\mathscr{D}\left(t\right)^{\ast} z_{1},\mathscr{D}\left(t\right)^{\ast} z_{2}\right\rangle ,\quad t_{0} < t < t_{1},\\{\mathscr{S}}\left(t_{0}\right)&={\mathscr{S}}_{0}.\nonumber \end{align} (2.12)To verify this, notice that if we suppose $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ is the generator of a strong evolution family on Z, then the function $$t\mapsto \left\langle{\mathscr{S}}\left (t\right )z_{1},z_{2}\right\rangle $$ may be differentiated on $$\left (t_{0},t_{1}\right )$$ to yield (2.12) (using the closedness of $$\mathscr{A}+\mathscr{B}\left (t\right )$$). So in essence what we have done is to reduce our stochastic system, consisting of (2.1) and (2.2) and the state estimator (2.5), to the completely deterministic system (2.12). We now express the cost functional (2.4) in a form which permits us to take advantage of (2.12). For this we set   $$ \mathscr{Q}=\left(\begin{array}{@{}cc@{}} Q & Q \\ Q & Q \end{array}\right),\quad\mathscr{R}=\left(\begin{array}{@{}cc@{}} R & 0 \\ 0 & 0 \end{array}\right). $$We may then meaningfully rewrite the cost functional to read (with an obvious change of notation from $${J}\left ({{u}}\left (\;\!\cdot \;\!\right )\right )$$ to $${J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$)   $$ {J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}E\,\big\langle \mathscr{Q}{{z}}\left(t_{1}\right),{{z}}\left(t_{1}\right)\!\big\rangle +\frac{1}{2}\int_{t_{0}}^{t_{1}}E\,\big\langle \! \left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){{z}}\left(t\right),{{z}}\left(t\right)\!\big\rangle \ \mathrm{d}t, $$satisfying the requirement in the sense of (2.6). Further, recalling that $$\mathscr{S}\left (t\right )$$ is the covariance operator of $${{z}}\left (t\right )$$, we can then write   \begin{align} {J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}\operatorname{Tr}\mathscr{Q}\mathscr{S}\left(t_{1}\right)+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right)\mathscr{S}\left(t\right)\,\mathrm{d}t, \end{align} (2.13)wherein Tr denotes the trace. It is not difficult to see—remember that we assumed earlier that Q was nuclear—that the operators $$\mathscr{Q}\mathscr{S}\left (t\right )$$ and $$\mathscr{K}\left (t\right )^{\ast }\mathscr{R}\mathscr{K}\left (t\right )\mathscr{S}\left (t\right )$$ are nuclear for $$t_{0}\leqslant t\leqslant t_{1}$$ even though $$\mathscr{S}\left (t\right )$$ is not, which implies that the expression on the right-hand side of (2.13) makes sense. (Notice that if we do not assume explicitly that the covariance operator $$S_{0}$$ is nuclear—and this we did not do—so that $$\mathscr{S}_{0}$$ is, then we cannot conclude from (2.11) that $$\mathscr{S}\left (t\right )$$ is nuclear for $$t_{0}\leqslant t\leqslant t_{1}$$.) To summarize, we have recast the stochastic control problem consisting of (2.1), (2.2) and (2.4) into the deterministic problem of finding the optimal pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$ for (2.12) in the product space $${\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (X,\mathbb{R}^{m}\right )\right )\oplus{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (\mathbb{R}^{p},X\right )\right )$$ as to minimize the cost functional (2.13). Henceforth, we shall refer to the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$ as the (possibly only locally) minimizing pair of feedback and estimator gain operators, and for notational convenience we write   $$ \mathcal{P}:={\boldsymbol{P}}_{\infty}(\left[t_{0},t_{1}\right];\mathcal{L}(X,\mathbb{R}^{m}))\oplus{\boldsymbol{P}}_{\infty}(\left[t_{0},t_{1}\right];\mathcal{L}(\mathbb{R}^{p},X)). $$ 3. Necessary conditions and reduction to the separation principle In this section we set out to solve the minimization problem posed in Section 2 by first deriving a set of necessary conditions for $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ to provide an extremum for the cost functional as given by (2.13) subject to (2.12). Recalling that the problem is now an entirely deterministic one, this is done in a most natural way with the aid of the variational calculus of operators, in particular, using the existence of the Gateaux differential. We will then on closing this section remark, without proof, that by the (assumed) existence of the minimizing pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\ \textrm{in}\ \mathcal{P}$$ the local minimum found is, in fact, the unique minimum. Let us turn briefly to the existence question. At this point, before entering into the discussion of necessary conditions, we would have to prove that the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$ actually exists in $$\mathcal{P}$$ or, at least, some (appropriate) subset thereof. Here we move directly to the question of necessary conditions, leaving the consideration of this particular synthesis problem to some future occasion. A brief outline of how such a proof might be carried out shall nevertheless be attempted. If one were to admit the possibility of a pair $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ being restrained to some subset of $$\mathcal{P}$$, one would have to begin by establishing weak-star compactness of bounded subsets of that space. The weak-star topology for $$\mathcal{P}$$ can be considered for this. In our case of unrestrained $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$, however, given the boundedness of any minimizing sequence of the pair in the space $$\mathcal{P}$$, the convergence of a subsequence in the weak-star topology is guaranteed in $$\mathcal{P}$$. Thus, using the immediately verified fact that the cost functional, as defined by (2.13), is positive, all that remains is to show that the map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ is lower semicontinuous on $$\mathcal{P}$$ in the weak-star topology. This by itself is an intricate matter, however, resulting in a rather lengthy proof (see, e.g. the paper by Ahmed, 2015), which we shall not go into here. For all the work in this section, we assume $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ attains a minimum on $$\mathcal{P}$$, i.e. there is a minimizing pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$. We begin with two lemmas which will be important later in the proof of our necessary conditions. Their proofs are elementary and are, therefore, omitted. For clarity and to avoid confusion in some parts of what follows, with some abuse of notation we sometimes write the covariance operator $${\mathscr{S}}\left (t\right )$$ as $${\mathscr{S}}\left (K\left (t\right ),L\left (t\right )\right )$$ to highlight its dependence on the feedback and estimator gain operators. To this end, we write $${\mathscr{S}}\left (K\left (t\right )\right )$$ or $${\mathscr{S}}\left (L\left (t\right )\right )$$ in place of $${\mathscr{S}}\left (K\left (t\right ),L\left (t\right )\right )$$, depending on whether we are to evaluate the Gateaux (partial) differential of the map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{\mathscr{S}}\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ at $$K\left (\;\!\cdot \;\!\right )$$ or $$L\left (\;\!\cdot \;\!\right )$$. In like vein, we also sometimes write $${J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$ or $${J}\left ({{L}}\left (\;\!\cdot \;\!\right )\right )$$ in place of $${J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$. In order to calculate the Gateaux differentials it will be necessary to introduce the operators $$F\left (t\right ):X\rightarrow \mathbb{R}^{m}$$ and $$G\left (t\right ):\mathbb{R}^{p}\rightarrow X$$ where $$F\left (t\right )\in \mathcal{L}\left (X,\mathbb{R}^{m}\right )$$ and $$G\left (t\right )\in \mathcal{L}\left (\mathbb{R}^{p},X\right )$$ for each t, and the maps $$t\mapsto F\left (t\right )$$ and $$t\mapsto G\left (t\right )$$ are both assumed to be strongly continuous on $$\left [t_{0},t_{1}\right ]$$. Set   $$ \mathscr{F}\left(t\right)=\left(\begin{array}{@{}cc@{}} F\left(t\right) & 0 \\ 0 & 0 \end{array}\right)\quad\textrm{and}\quad\mathscr{G}\left(t\right)=\left(\begin{array}{@{}cc@{}} 0 & G\left(t\right) \\ 0 & -G\left(t\right) \end{array}\right). $$ Lemma 3.1 Consider the system (2.12). The map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{\mathscr{S}}\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ is Gateaux differentiable on $$\mathcal{P}$$, and the following assertions hold: (i) the Gateaux differential of $$K\left (\;\!\cdot \;\!\right )\mapsto{\mathscr{S}}\left (K\left (\;\!\cdot \;\!\right )\right )$$ is given by the solution for $$t_{0}\leqslant t\leqslant t_{1}$$ of   \begin{align*} \left\langle \dot{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{1},z_{2}\right\rangle &=\left\langle{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \left(\mathscr{A}\!+\!\mathscr{B}\left(t\right)\right)^{\ast} z_{1},{\mathscr{S}}^{\prime}\!\left(K\!\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \!+\!\left\langle{\mathscr{S}}\left(K\!\left(t\right)\right)z_{1},\mathscr{F}\left(t\right)^{\ast} \mathscr{B}^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \mathscr{F}\left(t\right)^{\ast} \mathscr{B}^{\ast} z_{1},{\mathscr{S}}\left(K\left(t\right)\right) z_{2}\right\rangle, \\{\mathscr{S}}^{\prime}\left(K\left(t_{0}\right);F\left(t_{0}\right)\right)&=0; \end{align*} (ii) the Gateaux differential of $$L\left (\;\!\cdot \;\!\right )\mapsto{\mathscr{S}}\left (L\left (\;\!\cdot \;\!\right )\right )$$ is given by the solution for $$t_{0}\leqslant t\leqslant t_{1}$$ of   \begin{align*} \left\langle \dot{\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)z_{1},z_{2}\right\rangle &=\left\langle{\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{1},{\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)z_{2}\right\rangle \!+\!\left\langle{\mathscr{S}}\left(L\left(t\right)\right)z_{1},\mathscr{C}^{\ast} \mathscr{G}\left(t\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \mathscr{C}^{\ast} \mathscr{G}\left(t\right)^{\ast} z_{1},{\mathscr{S}}\left(L\left(t\right)\right) z_{2}\right\rangle +\left\langle \mathscr{N}\mathscr{G}\left(t\right)^{\ast} z_{1},\mathscr{L}\left(t\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \mathscr{L}\left(t\right)^{\ast} z_{1},\mathscr{N}\mathscr{G}\left(t\right)^{\ast} z_{2}\right\rangle, \\{\mathscr{S}}^{\prime}\left(L\left(t_{0}\right);G\left(t_{0}\right)\right)&=0. \end{align*} Lemma 3.2 Consider the cost functional (2.13). The map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ is Gateaux differentiable on $$\mathcal{P}$$, and the following assertions hold: (i) the Gateaux differential of $$K\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$ is given by   \begin{align*} {J}^{\prime}\!\left({{K}}\!\left(\;\!\cdot\;\!\right)\!;{{F\!}}\left(\;\!\cdot\;\!\right)\right)\!&=\!\frac{1}{2}\operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\!\left(K\!\left(t_{1}\right)\!;F\!\left(t_{1}\right)\right)\\ &\quad+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left[\left(\mathscr{Q}\!+\!\mathscr{K}\!\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\!\left(K\!\left(t\right);F\left(t\right)\right)\right.\\ &\quad\left.+\left(\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{F}\left(t\right)+\mathscr{F}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}\left(K\left(t\right)\right)\right]\,\mathrm{d}t; \end{align*} (ii) the Gateaux differential of $$L\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{L}}\left (\;\!\cdot \;\!\right )\right )$$ is given by   \begin{align*} {J}^{\prime}\left({{L}}\left(\;\!\cdot\;\!\right);{{G}}\left(\;\!\cdot\;\!\right)\right)&=\frac{1}{2}\operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\left(L\left(t_{1}\right);G\left(t_{1}\right)\right)\\[-2pt]&\quad+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)\,\mathrm{d}t. \end{align*} The results in Lemma 3.2, as they are at the moment, are difficult to evaluate and do not help us establish necessary conditions in a form that we shall find useful. The source of the difficulty is the appearance of $${\mathscr{S}}^{\prime }\left (K\left (\;\!\cdot \;\!\right );F\left (\;\!\cdot \;\!\right )\right )$$ and $${\mathscr{S}}^{\prime }\left (L\left (\;\!\cdot \;\!\right );G\left (\;\!\cdot \;\!\right )\right )$$, respectively, in the expressions for $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$ and $${J}^{\prime }\left ({{L}}\left (\;\!\cdot \;\!\right );{{G}}\left (\;\!\cdot \;\!\right )\right )$$. Still, they will play a crucial role in the following, and we shall see presently that a very simple manipulation provides a way round this problem. Before we do this, however, we make use of a standard result to state the following lemma, thereby introducing the so-called ‘adjoint’ system. Lemma 3.3 For $$z_{1},z_{2}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ the function $$\left\langle{\mathscr{P}}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ exists and is the unique selfadjoint solution of the system   \begin{align} \left\langle \dot{\mathscr{P}}\left(t\right)z_{1},z_{2}\right\rangle &=-\left\langle{\mathscr{P}}\left(t\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right) z_{2}\right\rangle -\left\langle \left(\mathscr{A}+\mathscr{B}\left(t\right)\right) z_{1},{\mathscr{P}}\left(t\right)z_{2}\right\rangle \nonumber\\ &\quad-\left\langle \mathscr{Q}z_{1},z_{2}\right\rangle -\left\langle \mathscr{R}\mathscr{K}\left(t\right)z_{1},\mathscr{K}\left(t\right)z_{2}\right\rangle ,\quad \quad t_{0}< t< t_{1},\\\nonumber\quad{\mathscr{P}}\left(t_{1}\right)&=\mathscr{Q}. \end{align} (3.1)Furthermore, $${\mathscr{P}}\left (t\right )\in \mathcal{L}\left (Z\right )$$, which is positive and nuclear for each fixed value of t, and $${\mathscr{P}}\left (\;\!\cdot \;\!\right )z_{1}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$. Proof We sketch the proof. The integrated form of (3.1) is, for any $$z_{1}\in Z$$,   $$ \mathscr{P}\left(t\right)z_{1}=\mathscr{U}\left(t_{1},t\right)^{\ast}\mathscr{Q}\mathscr{U}\left(t_{1},t\right) z_{1}+\int^{t_{1}}_{t}\mathscr{U}\left({s},t\right)^{\ast}\left(\mathscr{Q}+\mathscr{K}\left({s}\right)^{\ast}\mathscr{R}\mathscr{K}\left({s}\right)\right)\mathscr{U}\left({s},t\right) z_{1}\,\,\mathrm{d}{s}. $$We know that $$\mathscr{Q}\in \mathcal{L}\left (Z\right )$$ and is positive symmetric. In the preceding section we have assumed implicitly that $$K\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (X,\mathbb{R}^{m}\right )\right )$$ and $$L\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (\mathbb{R}^{p},X\right )\right )$$, whence $$\mathscr{K}\left (t\right )^{\ast }\mathscr{R}\mathscr{K}\left (t\right )\in \mathcal{L}\left (Z\right )$$. Thus, $${\mathscr{P}}\left (t\right )\in \mathcal{L}\left (Z\right )$$ and is, for each t, positive symmetric, and hence selfadjoint. In fact, in keeping with the remarks following (2.11), we see immediately that $${\mathscr{P}}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (Z\right )\right )$$. Further, it follows from the nuclearity of $$\mathscr{Q}$$ that $${\mathscr{P}}\left (t\right )$$ is nuclear for $$t_{0}\leqslant t\leqslant t_{1}$$. The existence of a unique strongly continuous solution of (3.1) can be shown as follows. We first note that we need not assume here (as we do in the case of (2.11)) that $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ is the generator of a strong evolution family on Z, because the function $$t\mapsto \left\langle{\mathscr{P}}\left (t\right )z_{1},z_{2}\right\rangle $$ must by definition be differentiable on $$\left (t_{0},t_{1}\right )$$ to give (3.1). This requires, however, that we proceed somewhat indirectly when it comes to proving uniqueness, by exploiting the density of $$\boldsymbol{D}\left (\mathscr{A}\right )$$ in Z. Let us suppose that $$\mathscr{B}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );\mathcal{L}\left (Z\right )\right )$$, so that the evolution family is strong. We may suppose further for the moment that $$\mathscr{D}\left (\;\!\cdot \;\!\right ){{r}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );Z\right )$$. Then for $$z_{0}\in \boldsymbol{D}\left (\mathscr{A}\right )$$, we may differentiate the function $$t\mapsto \left\langle \left ({\mathscr{P}}\left (t\right )-{\mathscr{M}}\left (t\right )\right ){{z}}\left (t\right ),{{z}}\left (t\right )\right\rangle $$, where $$\left\langle{\mathscr{M}}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ for $$z_{1},z_{2}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ is another solution of (3.1), to get $${\mathscr{P}}\left (\;\!\cdot \;\!\right )={\mathscr{M}}\left (\;\!\cdot \;\!\right )$$. The results so far provide the required ingredients for the following development of necessary conditions. Proposition 3.1 Necessary conditions for a pair $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ to extremize $${J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ as defined by (2.13) are that, for $$t_{0}\leqslant t\leqslant t_{1}$$,   \begin{align} \operatorname{Tr}\mathscr{F}\!\left(t\right)^{\ast}\left(\mathscr{B}^{\ast}{\mathscr{P}}\!\left(t\right)+\mathscr{R}\mathscr{K}\!\left(t\right)\right){\mathscr{S}}\!\left(t\right)\!=\!0\quad\textrm{and}\quad\operatorname{Tr}{\mathscr{P}}\!\left(t\right)\left({\mathscr{S}}\!\left(t\right)\mathscr{C}^{\ast}+\mathscr{L}\!\left(t\right)\mathscr{N}\right)\mathscr{G}\left(t\right)^{\ast}\!=\!0, \end{align} (3.2)where for $$z_{1},z_{2}$$ in $$\boldsymbol{D}\left (\mathscr{A}\right )$$ or $$\boldsymbol{D}\left (\mathscr{A}^{\ast }\right )$$, as appropriate, the functions $$\left\langle \mathscr{P}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ and $$\left\langle \mathscr{S}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ satisfy (3.1) and (2.12), respectively. Proof We only prove the first necessary condition in (3.2) as the other is proven in a similar manner. Recall from Lemma 3.2(i) the expression for $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$. To begin with, we first of all show how to eliminate $${\mathscr{S}}^{\prime }\left (K\left (\;\!\cdot \;\!\right );F\left (\;\!\cdot \;\!\right )\right )$$ there, and in order to do so we compute formally, by using the adjoint system in Lemma 3.3, that   \begin{align*} \frac{{d}}{{d}t}&\left\langle{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)z_{1},z_{2}\right\rangle\\ &=\frac{{d}}{{d}t}\left\langle{\mathscr{P}}\left(t\right)z_{1},{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \\ &=-\left\langle \left[\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast}{\mathscr{P}}\left(t\right)+{\mathscr{P}}\left(t\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)+\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right]z_{1},{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \\ &\quad+\left\langle{\mathscr{P}}\left(t\right)z_{1},\dot{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \\ &=-\left\langle \left[{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast}{\mathscr{P}}\left(t\right)+{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)\right.\right. \\ &\qquad\quad-\dot{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)\!\!\left.\left.\vphantom{\big[}\right]z_{1},z_{2}\right\rangle -\left\langle{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right)z_{1},z_{2}\right\rangle \end{align*}for $$z_{1}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ and $$z_{2}\in \boldsymbol{D}\left (\mathscr{A}^{\ast }\right )$$. Then since all terms are integrable,   \begin{align*} &\left\langle{\mathscr{S}}^{\prime}\left(K\left(t_{1}\right);F\left(t_{1}\right)\right)\mathscr{Q}z_{1},z_{2}\right\rangle \\ &\quad=-\int^{t_{1}}_{t_{0}}\left\langle \left[{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast}{\mathscr{P}}\left(t\right)+{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)\right. \right. \\ &\qquad\qquad\qquad-\dot{\mathscr{S}}^{\prime}\!\left(K\!\left(t\right)\!;F\left(t\right)\right)\!{\mathscr{P}}\!\left(t\right)\!\!\left.\left.\vphantom{\big[}\right]z_{1},z_{2}\right\rangle \,\mathrm{\!d}t -\!\int^{t_{1}}_{t_{0}}\!\left\langle{\mathscr{S}}^{\prime}\!\left(K\!\left(t\right);F\!\left(t\right)\right)\left(\mathscr{Q}\!+\!\mathscr{K}\!\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\!\left(t\right)\right)z_{1},z_{2}\right\rangle \,\mathrm{d}t, \end{align*}where we have taken into account the initial and end conditions (see Lemma 3.1(i) and Lemma 3.3)   $$ {\mathscr{S}}^{\prime}\left(K\left(t_{0}\right);F\left(t_{0}\right)\right)=0,\quad{\mathscr{P}}\left(t_{1}\right)=\mathscr{Q}. $$Taking the trace of both sides, we obtain in view of Lemma 3.1(i), and using the cyclic properties of the trace operation,   \begin{align*} \operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\left(K\left(t_{1}\right);F\left(t_{1}\right)\right)&=\int^{t_{1}}_{t_{0}}\operatorname{Tr}{\mathscr{P}}\left(t\right)\left(\mathscr{B}\mathscr{F}\left(t\right){\mathscr{S}}\left(K\left(t\right)\right)+{\mathscr{S}}\left(K\left(t\right)\right)\mathscr{F}\left(t\right)^{\ast}\mathscr{B}^{\ast}\right)\,\mathrm{d}t \\ &\quad-\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\,\mathrm{d}t. \end{align*}Thus, what we have shown is that   \begin{align*} \operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\left(K\left(t_{1}\right);F\left(t_{1}\right)\right)=&\int^{t_{1}}_{t_{0}}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Tr}{\mathscr{P}}\left(t\right){\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\,\mathrm{d}t \\ =&\int^{t_{1}}_{t_{0}}\operatorname{Tr}{\mathscr{P}}\left(t\right)\left(\mathscr{B}\mathscr{F}\left(t\right){\mathscr{S}}\left(K\left(t\right)\right)+{\mathscr{S}}\left(K\left(t\right)\right)\mathscr{F}\left(t\right)^{\ast}\mathscr{B}^{\ast}\right)\,\mathrm{d}t \\ &-\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\,\mathrm{d}t. \end{align*}Substituting the above relation into the original expression for $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$ now leads to   \begin{align*} {J}^{\prime}\left({{K}}\left(\;\!\cdot\;\!\right);{{F}}\left(\;\!\cdot\;\!\right)\right)&=\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}{\mathscr{P}}\left(t\right)\left(\mathscr{B}\mathscr{F}\left(t\right){\mathscr{S}}\left(K\left(t\right)\right)+{\mathscr{S}}\left(K\left(t\right)\right)\mathscr{F}\left(t\right)^{\ast}\mathscr{B}^{\ast}\right)\,\mathrm{d}t \\[-2pt] &\quad+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{F}\left(t\right)+\mathscr{F}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}\left(K\left(t\right)\right)\,\mathrm{d}t. \end{align*}All terms containing $${\mathscr{S}}^{\prime }\left (K\left (\;\!\cdot \;\!\right );F\left (\;\!\cdot \;\!\right )\right )$$ have been eliminated from the original $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$ at this point, and it immediately follows, on reordering the traces, that   $$ {J}^{\prime}\left({{K}}\left(\;\!\cdot\;\!\right);{{F}}\left(\;\!\cdot\;\!\right)\right)=\int^{t_{1}}_{t_{0}}\operatorname{Tr}\mathscr{F}\left(t\right)^{\ast}\left(\mathscr{B}^{\ast}{\mathscr{P}}\left(t\right)+\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}\left(K\left(t\right)\right)\,\mathrm{d}t. $$In order that $$K\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (X,\mathbb{R}^{m}\right )\right )$$ furnish an extremum of the map $$K\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$, it is necessary that $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )=0$$, and we clearly have the first necessary condition in (3.2), completing the first part of the proposition. The result of Proposition 3.1 enables us to obtain—without using Hilbert space probability theory—an alternative generalization of the separation principle to encompass infinite-dimensional systems. This is our main result, which is embodied in the next theorem and its corollary; but before we go into the statement and proof of the theorem, it will be necessary to decompose the operators $${\mathscr{P}}\left (t\right )\in \mathcal{L}\left (Z\right )$$ and $${\mathscr{S}}\left (t\right )\in \mathcal{L}\left (Z\right )$$ according to the decompositions   $$ {\mathscr{P}}\left(t\right)=\left(\begin{array}{@{}cc@{}} P_{11}\left(t\right) & P_{12}\left(t\right) \\ P_{21}\left(t\right) & P_{22}\left(t\right) \end{array}\right)\quad\textrm{and}\quad{\mathscr{S}}\left(t\right)=\left(\begin{array}{@{}cc@{}} S_{11}\left(t\right) & S_{12}\left(t\right) \\ S_{21}\left(t\right) & S_{22}\left(t\right) \end{array}\right). $$Since $${\mathscr{P}}\left (t\right )$$ and $${\mathscr{S}}\left (t\right )$$ are both symmetric (which follows from our remarks following the statement of (2.11), and Lemma 3.3), it holds that   $$ P_{21}\left(t\right)=P_{12}\left(t\right)^{\ast}\quad\textrm{and}\quad S_{21}\left(t\right)=S_{12}\left(t\right)^{\ast}. $$ Theorem 3.1 Consider the system (2.12), and let the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ minimize the cost functional defined by (2.13). Then we have, for $$t_{0}\leqslant t\leqslant t_{1}$$, that   \begin{align} K^{o}\left(t\right)=-R^{-1}B^{\ast} P_{11}\left(t\right)\quad\textrm{and}\quad L^{o}\left(t\right)=S_{22}\left(t\right)C^{\ast} W^{-1}, \end{align} (3.3)where for $$x_{1},x_{2}$$ in $$\boldsymbol{D}\left ({A}\right )$$ or $$\boldsymbol{D}\left ({A}^{\ast }\right )$$, as appropriate, the functions $$\left\langle P_{11}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ and $$\left\langle S_{22}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ exist and are, respectively, the unique selfadjoint solutions for $$t_{0}\leqslant t\leqslant t_{1}$$ of   \begin{align*} \left\langle \dot{P}_{11}\left(t\right)x_{1},x_{2}\right\rangle &=-\big\langle{P}_{11}\left(t\right)x_{1},{A} x_{2}\big\rangle -\big\langle{A} x_{1},{P}_{11}\left(t\right)x_{2}\big\rangle -\big\langle Qx_{1},x_{2}\big\rangle +\big\langle{R}^{-1}B^{\ast}{P}_{11}\left(t\right) x_{1},B^{\ast} {P}_{11}\left(t\right) x_{2}\big\rangle ,\\[-2pt] {P}_{11}\left(t_{1}\right)&=Q, \end{align*}and   \begin{align*} \left\langle \dot{S}_{22}\left(t\right)x_{1},x_{2}\right\rangle &=\left\langle{S}_{22}\left(t\right)x_{1},{A}^{\ast} x_{2}\right\rangle +\left\langle{A}^{\ast} x_{1},{S}_{22}\left(t\right)x_{2}\right\rangle +\left\langle VD^{\ast} x_{1},D^{\ast} x_{2}\right\rangle -\big\langle W^{-1} C{S}_{22}\left(t\right) x_{1},C{S}_{22}\left(t\right)x_{2}\big\rangle , \\[-2pt]{S}_{22}\left(t_{0}\right)&=S_{0}. \end{align*} Remark 3.1 As we shall see in the proof of the theorem (and as is already apparent), the differential operator equations in the theorem are the inner product versions of the well-known Riccati equations of the finite-dimensional LQG theory. These equations have been studied extensively (the earliest studies, by Kalman, 1960 and Wonham, 1968a, go back to the beginnings of optimal control theory in the 1960s), and many properties such as existence, uniqueness, continuity and asymptotic behaviour of their solutions are known. For the generalization of the finite dimensional results to the infinite dimensional case of interest here, see the books by Curtain & Pritchard (1978) and Balakrishnan (1981) or, to some extent, the book by Curtain & Zwart (1995). Proof of Theorem 3.1 We deduced in Proposition 3.1 that in seeking the extrema of the maps $$K\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$ and $$L\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{L}}\left (\;\!\cdot \;\!\right )\right )$$ simultaneously over $$\mathcal{P}$$ it is necessary that the conditions in (3.2) hold. Using the partitioned elements for $${\mathscr{P}}\left (t\right )$$ and $${\mathscr{S}}\left (t\right )$$ in (3.2), it may be seen that, since the pair $$\left (F\left (\;\!\cdot \;\!\right ),G\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ is arbitrary, this will be achieved if and only if for $$t_{0}\leqslant t\leqslant t_{1}$$ we have that   \begin{align} (B^{\ast}{P}_{11}(t)+RK(t)){S}_{11}(t)+B^{\ast}{P}_{12}(t){S}_{12}(t)^{\ast}=0 \end{align} (3.4)and   \begin{align} ({P}_{11}(t)-{P}_{12}(t)^{\ast})({S}_{12}(t)C^{\ast} +L(t) W)+({P}_{12}(t)-{P}_{22}(t))({S}_{22}(t)C^{\ast}-L(t) W)=0. \end{align} (3.5)We find that making use of the partitioned expressions for $${\mathscr{P}}\left (t\right )$$ and $${\mathscr{S}}\left (t\right )$$ in (3.1) and (2.12) will yield the decomposition into, in each case, four coupled differential operator equations. In view then of the fact (this can be verified by routine computations) that   \begin{align} P_{11}\left(\;\!\cdot\;\!\right)=P_{12}\left(\;\!\cdot\;\!\right)^{\ast}\quad\textrm{and}\quad S_{12}\left(\;\!\cdot\;\!\right)=0, \end{align} (3.6)we obtain from the two relations (3.4) and (3.5) the expressions for the gain operators in (3.3) together with the resulting problems in the statement of the theorem for the Riccati equations (which are simply the appropriate partitionings of (3.1) and (2.12) with the expressions in (3.3) substituted in them). The existence and uniqueness of the functions $$\left\langle P_{11}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ and $$\left\langle S_{22}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ are standard results (Balakrishnan, 1981, Sections 5.2 and 6.8); refer also to the proof of Lemma 3.3. The fact that $${P}_{11}\left (t\right )$$ and $${S}_{22}\left (t\right )$$ are selfadjoint is obvious. What remains to be verified is that the joint extremum found is, in fact, a minimum. To begin with, we note, first of all, that the quantities $$\operatorname{Tr}\mathscr{Q}\mathscr{S}\left (t_{1}\right )$$ and $$\operatorname{Tr}\left (\mathscr{Q}+\mathscr{K}\left (t\right )^{\ast }\mathscr{R}\mathscr{K}\left (t\right )\right )\mathscr{S}\left (t\right )$$ which appear in the cost functional (2.13) can be partitioned as   $$ \operatorname{Tr}\mathscr{Q}\mathscr{S}\left(t_{1}\right)=\operatorname{Tr}QS_{11}\left(t_{1}\right)+\operatorname{Tr}QS_{22}\left(t_{1}\right) $$and   $$ \operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right)\mathscr{S}\left(t\right)=\operatorname{Tr}\left(Q+ K\left(t\right)^{\ast} RK\left(t\right)\right)S_{11}\left(t\right)+\operatorname{Tr}QS_{22}\left(t\right), $$where, in both, we have taken account of the latter condition in (3.6); and so,   \begin{align} {J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right)&=\frac{1}{2}\operatorname{Tr}QS_{11}\left(t_{1}\right)+\frac{1}{2}\operatorname{Tr}QS_{22}\left(t_{1}\right)\nonumber\\[-2pt] &\quad+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}\left(Q+K\left(t\right)^{\ast} RK\left(t\right)\right)S_{11}\left(t\right)\,\mathrm{d}t +\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}QS_{22}\left(t\right)\,\mathrm{d}t. \end{align} (3.7)It follows then from calculations similar to those made in the proof of Proposition 3.1, and using the corresponding Riccati equations in the statement of the theorem, that   \begin{align*} \operatorname{Tr}QS_{11}\left(t_{1}\right)&=\int^{t_{1}}_{t_{0}}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Tr}P_{11}\left(t\right)S_{11}\left(t\right)\,\mathrm{d}t \\[-2pt] &=\int^{t_{1}}_{t_{0}}\operatorname{Tr}{P}_{11}\left(t\right)S_{22}\left(t\right)C^{\ast} W^{-1}CS_{22}\left(t\right)\,\mathrm{d}t\!-\!\int^{t_{1}}_{t_{0}}\operatorname{Tr}\big(Q+ P_{11}\left(t\right)BR^{-1}B^{\ast} P_{11}\left(t\right)\big)S_{11}\left(t\right)\,\mathrm{d}t. \end{align*}With this, and rearranging, the right-hand side of (3.7) becomes   \begin{multline*} \frac{1}{2}\operatorname{Tr}QS_{22}\left(t_{1}\right)+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}QS_{22}\left(t\right)\,\mathrm{d}t+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}{P}_{11}\left(t\right)S_{22}\left(t\right)C^{\ast} W^{-1}CS_{22}\left(t\right)\,\mathrm{d}t \\[-2pt] +\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}\big(K\left(t\right)^{\ast} RK\left(t\right)-P_{11}\left(t\right)BR^{-1}B^{\ast} P_{11}\left(t\right)\big)S_{11}\left(t\right)\,\mathrm{d}t, \end{multline*}and it is readily seen that $${J}\left ({{K}}^{o}\left (\;\!\cdot \;\!\right ),{{L}}^{o}\left (\;\!\cdot \;\!\right )\right )\leqslant{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$. This completes the proof. If we had proven the existence of the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$, we would have shown that the map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ is lower semicontinuous (in the weak-star sense) on $$\mathcal{P}$$. Assuming this result, as we did in the theorem, the fact that then the pair is unique lends to the following strengthening of Theorem 3.1. Corollary 3.1 The minimizing pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ is unique and is given by the individual expressions in (3.3). Then   \begin{align} {J}\left({{K}}^{o}\left(\;\!\cdot\;\!\right),{{L}}^{o}\left(\;\!\cdot\;\!\right)\right)\leqslant{J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right) \end{align} (3.8)for all $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$, and the minimal cost, realized with the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$, is given by   $$ {J}\left({{K}}^{o}\left(\;\!\cdot\;\!\right),{{L}}^{o}\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}\operatorname{Tr}QS_{22}\left(t_{1}\right)+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}QS_{22}\left(t\right)\,\mathrm{d}t+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}{P}_{11}\left(t\right)S_{22}\left(t\right)C^{\ast} W^{-1}CS_{22}\left(t\right)\,\mathrm{d}t. $$ 4. Concluding remarks In this paper an approach different from the common probabilistic ones has been considered to derive a solution to the infinite-dimensional LQG regulator problem. By transforming the problem into a deterministic framework we were able to derive, using standard variational arguments, a set of necessary conditions for the cost functional involved to take on an extremum (in fact, a unique minimum). In particular, we obtained a new method of proof for the separation principle that, because we consider the minimization problem on an extended Hilbert state-space, completely avoids abstract probability theory. References Ahmed, N. U. & Li, P. ( 1991) Quadratic regulator theory and linear filtering under system constraints. IMA J. Math. Control Inf. , 8, 93-- 107. Google Scholar CrossRef Search ADS   Ahmed, N. U. ( 1998) Linear and Nonlinear Filtering for Scientists and Engineers . Singapore: World Scientific. Google Scholar CrossRef Search ADS   Ahmed, N. U. ( 2015) Infinite dimensional uncertain dynamic systems on Banach spaces and their optimal output feedback control. Discuss. Math. Differ. Incl. Control Optim ., 35, 65-- 87. Google Scholar CrossRef Search ADS   Balakrishnan, A. V. ( 1981) Applied Functional Analysis . New York: Springer. Curtain, R. F. & Pritchard, A. J. ( 1978) Infinite Dimensional Linear Systems Theory . Berlin: Springer. Google Scholar CrossRef Search ADS   Curtain, R. F. & Zwart, H. J. ( 1995) An Introduction to Infinite-Dimensional Linear Systems Theory . New York: Springer. Google Scholar CrossRef Search ADS   Kalman, R. E. ( 1960) Contributions to the theory of optimal control. Bol. Soc. Mat. Mex. , 5, 102-- 119. Phillips, R. S. ( 1953) Perturbation theory for semi-groups of linear operators. Trans. Am. Math. Soc. , 74, 199-- 221. Google Scholar CrossRef Search ADS   Wonham, W. M. ( 1968a) On a matrix Riccati equation of stochastic control. SIAM J. Control , 6, 681-- 697. Google Scholar CrossRef Search ADS   Wonham, W. M. ( 1968b) On the separation theorem of stochastic control. SIAM J. Control , 6, 312-- 326. Google Scholar CrossRef Search ADS   © 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

A new method for proving the separation principle for the infinite-dimensional LQG regulator problem

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Abstract

Abstract This paper gives an alternative perspective on the standard linear quadratic Gaussian regulator problem for infinite-dimensional state-space systems. We will show that when considered on an extended Hilbert state-space, the originally stochastic control problem can be phrased as a deterministic one. In this setting we obtain a new method of proof for the well-known separation principle which does not involve probabilistic considerations. 1. Introduction A frequent occurrence in practical distributed parameter systems, with finite-dimensional inputs and outputs, is for them to be controlled by deriving optimal controls (if they exist) for the system subject not only to additive stochastic disturbances, but also to partial observations of the state. This is commonly known as the ‘partially observable optimal control problem’. If the system to be controlled is linear and the cost quadratic, then the minimization called for in the stochastic control problem will be a problem that clearly falls within the scope of the standard linear quadratic Gaussian (LQG) regulator problem and, if solved successfully, will lead to the construction of an optimal state estimate feedback control law relative to the quadratic cost functional involved. At the heart of the solution to the LQG regulator problem lies the so-called separation principle (originally stated and proven by Wonham, 1968b for finite-dimensional systems) which—in both infinite and finite dimensions—is normally investigated using abstract probability theory. The results are, of course, very well known, and for infinite-dimensional systems a few approaches are available; see the texts by Curtain & Pritchard (1978), Balakrishnan (1981) and the relevant references cited there. In the present paper we will be calling for the proof of the separation principle with the motivation of avoiding a probabilistic treatment. Some work has been done along similar lines; yet, an early work which might be considered most closely related to ours is probably that of Ahmed & Li (1991) (see also the book by Ahmed, 1998). Although they use a similar technique in the finite-dimensional case to reformulate and solve the individual problems of state estimation and feedback control inherent in the LQG regulator problem, they do not explicitly consider the joint problem of state estimate feedback control. We shall see here, however, that the technical details required for this will only be slightly more complex, even for infinite-dimensional systems, when the controls are assumed from the beginning to be of the standard state estimate feedback type. In our developments in this paper, in Section 2 to be precise, the LQG regulator problem is posed not on the usual Hilbert state-space, but rather on an extended Hilbert space. This allows us to convert the controlled stochastic system—in fact, a suitable ‘augmentation’ of it—into a deterministic one and reformulate the cost functional so that the dependence of the new cost functional will be on the feedback and estimator gain operators (definitions in Section 2) rather than only on the control inputs. In so doing, the new minimization problem, as we shall see, becomes considerably simpler than the original one in that it will be amenable to treatment within the framework of the variational calculus of operators. So by the end of Section 2 we will be well prepared to examine the question of necessary conditions. In Section 3 our intention, broadly put, will be to derive a solution to our minimization problem. We will not be concerned with what is mathematically a not-so-trivial problem, namely, the problem of existence of the optimal pair of feedback and estimator gain operators in spaces of operator functions. Rather, we will derive a set of necessary conditions and thereby provide a different approach to the proof of the separation principle in infinite dimensions, which—we already touched upon this—does not rely on a probabilistic treatment. In point of fact, our main achievement there will be to establish the separation principle without employing the theory of Wiener processes and stochastic integration and hence to offer an alternative route to generalizing the LQG regulator problem to an infinite-dimensional setting. Our method of proof is new, and we believe that not only does it complement the direct approach as considered, for example, by Balakrishnan in his book, but is interesting for another reason: it has appealing mathematical simplicity because it relies almost exclusively on the variational arguments. The solution for the feedback and estimator gain operators entails two sets of four non-standard, coupled differential operator equations each, which can be shown by simple arguments to reduce to the well-known Riccati equations that arise in the solution to the LQG regulator problem. The final result in Section 3 then usefully indicates how an existence theorem for the control problem (which, as we have noted earlier, we do not present here) would imply the uniqueness of the optimal pair of feedback and estimator gain operators thus found. As a final note in closing, the approach presented in this paper differs, somewhat conceptually, from that of most of the literature in that the structure of the state estimator is fixed beforehand. The state estimator plays a central role in forming the augmented system, and this means in effect that the equations describing the state estimator become part of the control problem formulation, strictly speaking, and not the solution. 2. An equivalent deterministic form of the LQG regulator problem We will deal in this paper with systems of rather general type described by (strongly continuous) $$C_{0}$$-semigroups and evolution families, and we begin by briefly formulating the standard LQG regulator problem as it applies to linear, time-invariant, stochastic dynamical systems in an infinite-dimensional Hilbert space setting. We will use in this section (and in fact throughout the whole paper) some of the notation and results of the book by Curtain & Zwart (1995) to which we refer the reader for more details. Consider any time interval $$\left [t_{0},t_{1}\right ]\subset \mathbb{R}_{+}$$, with $$0\leqslant t_{0} < t_{1}$$, wherein the controlled system is to evolve. We start with consideration of the abstract system   \begin{align} \dot{x}\left(t\right)&=Ax\left(t\right)+Bu\left(t\right)+Dv\left(t\right),\quad t_{0} < t < t_{1},\nonumber\\[-3pt] \quad x\left(t_{0}\right)&=x_{0}, \end{align} (2.1)on some real separable Hilbert state-space X, accompanied by the output equation   \begin{align} y\left(t\right)=Cx\left(t\right)+w\left(t\right),\quad t_{0}\leqslant t\leqslant t_{1}. \end{align} (2.2)The system operator $$A:\boldsymbol{D}\left (A\right )\left (\subset X\right )\rightarrow X$$, which is (as is usually the case in physical dynamical systems) maximal dissipative, thus closed and densely defined, and hence is the infinitesimal generator of a $$C_{0}$$-semigroup $$T\left (\;\!\cdot \;\!\right ):\left [t_{0},t_{1}\right ]\rightarrow \mathcal{L}\left (X\right )$$ of contractions. For each fixed $$x_{0}\in X$$ one then has $$T\left (\;\!\cdot \;\!\right )x_{0}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];X\right )$$, i.e. the map $$t\mapsto T\left (t\right )$$ is continuous in the strong topology of $$\mathcal{L}\left (X\right )$$. We assume the mappings B, C and D are all finite-dimensional, in the sense that they have finite rank and are bounded, $$B,D\in \mathcal{L}\left (\mathbb{R}^{m},X\right )$$ and $$C\in \mathcal{L}\left (X,\mathbb{R}^{p}\right )$$. As usual, we will allow for discontinuous control inputs and thus assume $$u\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$. To ‘randomize’ the system in some sense, and in accordance with the standard LQG regulator problem statement, let it be assumed that the initial state $$x_{0}$$ is a Gaussian random variable, having zero mean, and covariance operator $$S_{0}\in \mathcal{L}\left (X\right )$$, which we will not assume to be necessarily nuclear. We will take into account the functions $$v\left (\;\!\cdot \;\!\right )$$ and $$w\left (\;\!\cdot \;\!\right )$$ as random entities also, which correspond to what is called ‘Gaussian white noise’ in the literature, but without having to resort to the theory of stochastic integration for Wiener processes. A rigorous justification for doing so rests with Balakrishnan’s formalism of white noise theory (in his book mentioned in the introduction) so full details are not necessary here. Suffice it to say that $$v\left (\;\!\cdot \;\!\right ),w\left (\;\!\cdot \;\!\right )$$ have sample paths which are norm square-integrable on $$\left [t_{0},t_{1}\right ]$$, i.e. they are, respectively, elements of $$\boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$ and $$\boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{p}\right )$$ and such that for any $$f\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$ and $$g\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{p}\right )$$ the quantities (where $$\left\langle \;\!\cdot \;\!,\;\!\cdot \;\!\right\rangle $$ denotes the usual Euclidean inner product)   $$ \int_{t_{0}}^{t_{1}} \left\langle v\left(t\right),f\left(t\right)\right\rangle \,\mathrm{d}t\quad\textrm{and}\quad\int_{t_{0}}^{t_{1}} \left\langle w\left(t\right),g\left(t\right)\right\rangle \,\mathrm{d}t $$are Gaussian with zero mean and positive symmetric covariance matrices. We take the matrices $$V\in \mathcal{L}\left (\mathbb{R}^{m}\right )$$ and $$W\in \mathcal{L}\left (\mathbb{R}^{p}\right )$$ as the covariances of $$v\left (\;\!\cdot \;\!\right )$$ and $$w\left (\;\!\cdot \;\!\right )$$, respectively, but to prove the direct analogue of the separation principle we will not, in general, consider the case where W is only positive. Rather, we shall require W to be bounded below (and hence boundedly invertible), so $$W^{-1}\in \mathcal{L}\left (\mathbb{R}^{p}\right )$$. This corresponds to the case of entirely non-perfect observations of the state. Further, and this is our final assumption, $$v\left (\;\!\cdot \;\!\right )$$, $$w\left (\;\!\cdot \;\!\right )$$ and $$x_{0}$$ are assumed mutually independent. The initial state $$x_{0}$$ can be any element of X, and because $$u\left (\;\!\cdot \;\!\right ),v\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2}\left (\left [t_{0},t_{1}\right ];\mathbb{R}^{m}\right )$$, we have only the weak or mild version of (2.1), which is written in terms of $$T\left (\;\!\cdot \;\!\right )$$,   \begin{align} x\left(t\right)=T\left(t-t_{0}\right)x_{0}+\int^{t}_{t_{0}}T\left(t-{s}\right)Bu\left({s}\right)\,\mathrm{d}{s}+\int^{t}_{t_{0}}T\left(t-{s}\right)Dv\left({s}\right)\,\mathrm{d}{s},\quad t_{0}\leqslant t\leqslant t_{1}. \end{align} (2.3)We therefore define the integral equation (2.3) to be the generalized solution of (2.1), which exists and is unique, and $$x\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];X\right )$$. Moreover, under our assumptions on B, D and $$x_{0}$$, the state $$x\left (t\right )$$ is a Gaussian random variable with zero mean for each fixed value of t. In the paper we shall almost always work with the differential notation (2.1), bearing in mind that what we really mean is the integral version (2.3). Suppose that the controls $$u\left (\;\!\cdot \;\!\right )$$ are ‘admissible’ if $$u\left (t\right )$$ is, in a sense that will be made more precise shortly, based on the output as defined by (2.2) up to time t for $$t_{0}\leqslant t\leqslant t_{1}$$. Given then the operator $$Q\in \mathcal{L}\left (X\right )$$, which is positive symmetric and nuclear, and some positive symmetric matrix $$R\in \mathcal{L}\left (\mathbb{R}^{m}\right )$$, bounded below so that $$R^{-1}\in \mathcal{L}\left (\mathbb{R}^{m}\right )$$, we associate with (2.3) the so-called cost of $$u\left (\;\!\cdot \;\!\right )$$, defined by   \begin{align} {J}\left(u\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}E\left\langle Qx\left(t_{1}\right),x\left(t_{1}\right)\right\rangle +\frac{1}{2}\int_{t_{0}}^{t_{1}}\left[E\left\langle Qx\left(t\right),x\left(t\right)\right\rangle + E\left\langle Ru\left(t\right),u\left(t\right)\right\rangle \right]\,\mathrm{d}t, \end{align} (2.4)wherein E denotes the expected value. (We have not distinguished in the notation above between either of the inner products on the spaces considered as this is clear from the context.) Let it, then, be required to find the admissible controls $$u\left (\;\!\cdot \;\!\right )$$ which minimize $${J}\left (u\left (\;\!\cdot \;\!\right )\right )$$. This is the well-known LQG regulator problem relative to an abstract semigroup formulation of the underlying system dynamics in Hilbert space. It is now possible to convert the stochastic control problem in a fairly straightforward way into an equivalent deterministic one. This can be done, as we now show, under the assumption that we are to develop a linear state estimate feedback form of the solution to the problem. Suppose to this end that we are given a state estimator, essentially of the Luenberger type, for the system consisting of (2.1) and (2.2):   \begin{align} \dot{\hat{x}}\left(t\right)&=A\hat{x}\left(t\right)+Bu\left(t\right)+L\left(t\right)(y\left(t\right)-C\hat{x}\left(t\right)),\quad t_{0} < t < t_{1},\nonumber\\[-2pt] \hat{x}\left(t_{0}\right)&=0, \end{align} (2.5)where the map $$t\mapsto L\left (t\right )$$ is strongly continuous from $$\left [t_{0},t_{1}\right ]$$ into $$\mathcal{L}\left (\mathbb{R}^{p},X\right )$$. The operator L(t) will be called the ‘estimator gain operator’. Now suppose that to each t is associated another operator $$K\left (t\right )\in \mathcal{L}\left (X,\mathbb{R}^{m}\right )$$, the map $$t\mapsto{K}\left (t\right )$$ being strongly continuous on $$\left [t_{0},t_{1}\right ]$$ also. The work throughout this paper is carried out under the assumption that the control $$u\left (t\right )$$ is always generated via the feedback law   \begin{align} u\left(t\right)=K\left(t\right)\hat{x}\left(t\right),\quad t_{0}\leqslant t\leqslant t_{1}; \end{align} (2.6)the operator K(t) is called the feedback gain operator. Notice that both K(t) and L(t) are unknown operators at this stage. If we now proceed formally, substituting for u(t) from (2.6) into (2.1), we obtain   $$ \dot{x}\left(t\right)=Ax\left(t\right)+BK\left(t\right)\hat{x}\left(t\right)+Dv\left(t\right). $$Similarly, by substituting (2.6) into (2.5) and using (2.2),   $$ \dot{\hat{x}}\left(t\right)=\left(A+BK\left(t\right)\right)\hat{x}\left(t\right)+L\left(t\right)C(x\left(t\right)-\hat{x}\left(t\right))+L\left(t\right)w\left(t\right). $$We then define the estimation error   $$ \tilde{x}\left(t\right):= x\left(t\right)-\hat{x}\left(t\right), $$in terms of which we arrive at   \begin{align} \dot{\tilde{x}}\left(t\right)&=\left(A-L\left(t\right)C\right)\tilde{x}\left(t\right)+Dv\left(t\right)-L\left(t\right)w\left(t\right),\quad t_{0} < t < t_{1},\nonumber\\[-2pt]\quad{\tilde{x}}\left(t_{0}\right)&={\tilde{x}}_{0}, \end{align} (2.7)where $${\tilde{x}}_{0}=x_{0}$$. What we now do is to introduce the product space   $$ Z:= X\oplus X $$as the new state space, namely, the augmented state space. Consider the vectors   $$ {{z}}\left(t\right):=\left(\begin{array}{@{}c@{}} \hat{{x}}\left(t\right) \\ \tilde{{x}}\left(t\right) \end{array}\right)\in Z\quad\textrm{and}\quad{{r}}\left(t\right):=\left(\begin{array}{@{}c@{}} {{v}}\left(t\right) \\{{w}}\left(t\right) \end{array}\right)\in\mathbb{R}^{m+p}, $$and define a ‘new’ system operator $$\mathscr{A}$$ by setting   $$ \mathscr{A}=\left(\begin{array}{@{}cc@{}} A & 0 \\ 0 & A \end{array}\right), $$with domain   \begin{align} \boldsymbol{D}\left(\mathscr{A}\right):=\boldsymbol{D}\left(A\right)\oplus\boldsymbol{D}\left(A\right). \end{align} (2.8)In doing so, as will be seen below, it is implicit that   \begin{align*} \mathscr{B}&=\left(\begin{array}{@{}cc@{}} B & 0 \\ 0 & 0 \end{array}\right),\quad\mathscr{C}=\left(\begin{array}{@{}cc@{}} 0 & 0 \\ 0 & C \end{array}\right),\quad\mathscr{D}=\left(\begin{array}{@{}cc@{}} 0 & 0 \\ D & 0 \end{array}\right),\\ \mathscr{K}\left(t\right)&=\left(\begin{array}{@{}cc@{}} K\left(t\right) & 0 \\ 0 & 0 \end{array}\right),\quad\mathscr{L}\left(t\right)=\left(\begin{array}{@{}cc@{}} 0 & L\left(t\right) \\ 0 & -L\left(t\right) \end{array}\right), \end{align*}and we will frequently make explicit use of these block operator matrices in what is to follow. We let $$\mathscr{B}\left (t\right ):Z\rightarrow Z$$ and $$\mathscr{D}\left (t\right ):\mathbb{R}^{m+p}\rightarrow Z$$ be given by   $$ \mathscr{B}\left(t\right):=\mathscr{B}\mathscr{K}\left(t\right)+\mathscr{L}\left(t\right)\mathscr{C}\quad\textrm{and}\quad\mathscr{D}\left(t\right):=\mathscr{D}+\mathscr{L}\left(t\right), $$where the maps $$t\mapsto \mathscr{B}\left (t\right )$$ and $$t\mapsto \mathscr{D}\left (t\right )$$ are, by virtue of the assumed continuity properties of $${K}\left (\;\!\cdot \;\!\right )$$ and $$L\left (\;\!\cdot \;\!\right )$$, strongly continuous from $$\left [t_{0},t_{1}\right ]$$ into, respectively, $$\mathcal{L}\left (Z\right )$$ and $$\mathcal{L} (\mathbb{R}^{m+p},Z )$$, and so it follows that $$\mathscr{B}\left (\;\!\cdot \;\!\right )$$ and $$\mathscr{D}\left (\;\!\cdot \;\!\right )$$ are uniformly bounded in norm on $$\left [t_{0},t_{1}\right ]$$—i.e. $$\mathscr{B}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (Z\right )\right )$$ and $$\mathscr{D}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty } (\left [t_{0},t_{1}\right ];\mathcal{L} (\mathbb{R}^{m+p},Z ) )$$. The above considerations enable us now to combine (2.5) and (2.7) to write the augmented system as   \begin{align} {\dot{{z}}}\left(t\right)&=\left(\mathscr{A}+\mathscr{B}\left(t\right)\right){{z}}\left(t\right)+\mathscr{D}\left(t\right){{r}}\left(t\right),\quad t_{0} < t < t_{1},\nonumber\\[-2pt] {{z}}\left(t_{0}\right)&=z_{0}, \end{align} (2.9)for the augmented state-space Z, where   $$ z_{0}:=\left(\begin{array}{@{}c@{}} 0 \\ \tilde{x}_{0} \end{array}\right)\in Z. $$Obviously the operator $$\mathscr{A}$$ is the generator of a $$C_{0}$$-semigroup of contractions on Z, and we notice that the operator $$\mathscr{A}+\mathscr{B}\left (t\right )$$, having domain   $$ \boldsymbol{D}\left(\mathscr{A}+\mathscr{B}\left(\;\!\cdot\;\!\right)\right)=\boldsymbol{D}\left(\mathscr{A}\right), $$independent of t, is a bounded linear perturbation of the unperturbed system operator $$\mathscr{A}$$ for each t. Thus, we have that $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ generates a weak evolution family $$\mathscr{U}\left (\;\!\cdot \;\!,\;\!\cdot \;\!\right ):\left \{\left (t,s\right ):t_{0}\leqslant s\leqslant t\leqslant t_{1}\right \}\rightarrow \mathcal{L}\left (Z\right )$$, which possesses the standard properties of a strong evolution family with an important exception (for our purposes at least) that whereas the map $$s\mapsto \mathscr{U}\left (t,{s}\right )$$ is strongly differentiable, the map $$t\mapsto \mathscr{U}\left (t,{s}\right )$$ is not, even though $$\mathscr{B}\left (\;\!\cdot \;\!\right )z_{0}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$. Since $$z_{0}\in Z$$ and $${{r}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{L}_{2} (\left [t_{0},t_{1}\right ];\mathbb{R}^{m+p} )$$, we therefore conclude that we are again only able to define a generalized solution of (2.9),   \begin{align} {{z}}\left(t\right)=\mathscr{U}\left(t,t_{0}\right)z_{0}+\int^{t}_{t_{0}}\mathscr{U}\left(t,{s}\right)\mathscr{D}\left({s}\right){{r}}\left({s}\right)\,\mathrm{d}{s},\quad t_{0}\leqslant t\leqslant t_{1}, \end{align} (2.10)where $${{z}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$. Then, as before, since $$z_{0}$$ is a Gaussian random variable with zero mean and $$\mathscr{D}\left (t\right )$$ has finite rank for $$t_{0}\leqslant t\leqslant t_{1}$$, the augmented state $${{z}}\left (t\right )$$ is a Gaussian random variable having zero mean for each fixed value of t. We reiterate, as an aside, regarding the interpretation of (2.10) that for $$z_{0}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ the generalized solution is the differentiable solution, in the strong sense, provided that $$\mathscr{B}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );\mathcal{L}\left (Z\right )\right )$$ and $$\mathscr{D}\left (\;\!\cdot \;\!\right ){{r}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );Z\right )$$, i.e. the maps $$t\mapsto \mathscr{B}\left (t\right )$$ and $$t\mapsto \mathscr{D}\left (t\right ){{r}}\left (t\right )$$ are strongly differentiable on $$\left (t_{0},t_{1}\right )$$ (Phillips, 1953). Then the evolution family generated by $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ is a strong evolution family, so the map $$t\mapsto \mathscr{U}\left (t,s\right )z_{0}$$ is differentiable for each $$z_{0}\in \boldsymbol{D}\left (\mathscr{A}\right )$$. This will prove to be important for our sequel. Now, write   $$ \mathscr{S}_{0}=\left(\begin{array}{@{}cc@{}} 0 & 0 \\ 0 & S_{0} \end{array}\right),\quad \mathscr{N}=\left(\begin{array}{@{}cc@{}} V & 0 \\ 0 & W \end{array}\right), $$and denote by $$\mathscr{S}\left (t\right )$$ the covariance operator associated with $${{z}}\left (t\right )$$. Then by way of some straightforward calculations with covariances (Balakrishnan, 1981, p. 317), using the independence of $$z_{0}$$ and $${{r}}\left (\;\!\cdot \;\!\right )$$, we have, for any $$z_{1}\in Z$$, that   \begin{align} \mathscr{S}\left(t\right)z_{1}=\mathscr{U}\left(t,t_{0}\right)\mathscr{S}_{0}\mathscr{U}\left(t,t_{0}\right)^{\ast} z_{1}+\int^{t}_{t_{0}}\mathscr{U}\left(t,{s}\right)\mathscr{D}\left({s}\right)\mathscr{N}\mathscr{D}\left({s}\right)^{\ast}\mathscr{U}\left(t,{s}\right)^{\ast} z_{1}\,\,\mathrm{d}{s}. \end{align} (2.11)Clearly, since $$\mathscr{S}_{0}$$ and $$\mathscr{N}$$ are both positive symmetric, $$\mathscr{S}\left (t\right )$$ is positive symmetric for each t. Moreover, by the uniform boundedness of the norm of the map $$\left (t,s\right )\mapsto \mathscr{U}\left (t,{s}\right )$$ on finite time intervals, we have $$\mathscr{S}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (Z\right )\right )$$. In fact, $${\mathscr{S}}\left (\;\!\cdot \;\!\right )z_{1}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$, and we note in particular that for $$z_{1},z_{2}\in \boldsymbol{D}\left (\mathscr{A}^{\ast }\right )$$ the function $$t\mapsto \left\langle{\mathscr{S}}\left (t\right )z_{1},z_{2}\right\rangle $$—defined as in (2.11)—can be differentiated to satisfy the system (wherein $$\left\langle \;\!\cdot \;\!,\;\!\cdot \;\!\right\rangle $$ now denotes the inner product on Z)   \begin{align} \left\langle \dot{\mathscr{S}}\left(t\right)z_{1},z_{2}\right\rangle &=\left\langle{\mathscr{S}}\left(t\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{2}\right\rangle +\left\langle \left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{1},{\mathscr{S}}\left(t\right)z_{2}\right\rangle\nonumber \\ &\quad+\left\langle \mathscr{N}\mathscr{D}\left(t\right)^{\ast} z_{1},\mathscr{D}\left(t\right)^{\ast} z_{2}\right\rangle ,\quad t_{0} < t < t_{1},\\{\mathscr{S}}\left(t_{0}\right)&={\mathscr{S}}_{0}.\nonumber \end{align} (2.12)To verify this, notice that if we suppose $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ is the generator of a strong evolution family on Z, then the function $$t\mapsto \left\langle{\mathscr{S}}\left (t\right )z_{1},z_{2}\right\rangle $$ may be differentiated on $$\left (t_{0},t_{1}\right )$$ to yield (2.12) (using the closedness of $$\mathscr{A}+\mathscr{B}\left (t\right )$$). So in essence what we have done is to reduce our stochastic system, consisting of (2.1) and (2.2) and the state estimator (2.5), to the completely deterministic system (2.12). We now express the cost functional (2.4) in a form which permits us to take advantage of (2.12). For this we set   $$ \mathscr{Q}=\left(\begin{array}{@{}cc@{}} Q & Q \\ Q & Q \end{array}\right),\quad\mathscr{R}=\left(\begin{array}{@{}cc@{}} R & 0 \\ 0 & 0 \end{array}\right). $$We may then meaningfully rewrite the cost functional to read (with an obvious change of notation from $${J}\left ({{u}}\left (\;\!\cdot \;\!\right )\right )$$ to $${J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$)   $$ {J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}E\,\big\langle \mathscr{Q}{{z}}\left(t_{1}\right),{{z}}\left(t_{1}\right)\!\big\rangle +\frac{1}{2}\int_{t_{0}}^{t_{1}}E\,\big\langle \! \left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){{z}}\left(t\right),{{z}}\left(t\right)\!\big\rangle \ \mathrm{d}t, $$satisfying the requirement in the sense of (2.6). Further, recalling that $$\mathscr{S}\left (t\right )$$ is the covariance operator of $${{z}}\left (t\right )$$, we can then write   \begin{align} {J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}\operatorname{Tr}\mathscr{Q}\mathscr{S}\left(t_{1}\right)+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right)\mathscr{S}\left(t\right)\,\mathrm{d}t, \end{align} (2.13)wherein Tr denotes the trace. It is not difficult to see—remember that we assumed earlier that Q was nuclear—that the operators $$\mathscr{Q}\mathscr{S}\left (t\right )$$ and $$\mathscr{K}\left (t\right )^{\ast }\mathscr{R}\mathscr{K}\left (t\right )\mathscr{S}\left (t\right )$$ are nuclear for $$t_{0}\leqslant t\leqslant t_{1}$$ even though $$\mathscr{S}\left (t\right )$$ is not, which implies that the expression on the right-hand side of (2.13) makes sense. (Notice that if we do not assume explicitly that the covariance operator $$S_{0}$$ is nuclear—and this we did not do—so that $$\mathscr{S}_{0}$$ is, then we cannot conclude from (2.11) that $$\mathscr{S}\left (t\right )$$ is nuclear for $$t_{0}\leqslant t\leqslant t_{1}$$.) To summarize, we have recast the stochastic control problem consisting of (2.1), (2.2) and (2.4) into the deterministic problem of finding the optimal pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$ for (2.12) in the product space $${\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (X,\mathbb{R}^{m}\right )\right )\oplus{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (\mathbb{R}^{p},X\right )\right )$$ as to minimize the cost functional (2.13). Henceforth, we shall refer to the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$ as the (possibly only locally) minimizing pair of feedback and estimator gain operators, and for notational convenience we write   $$ \mathcal{P}:={\boldsymbol{P}}_{\infty}(\left[t_{0},t_{1}\right];\mathcal{L}(X,\mathbb{R}^{m}))\oplus{\boldsymbol{P}}_{\infty}(\left[t_{0},t_{1}\right];\mathcal{L}(\mathbb{R}^{p},X)). $$ 3. Necessary conditions and reduction to the separation principle In this section we set out to solve the minimization problem posed in Section 2 by first deriving a set of necessary conditions for $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ to provide an extremum for the cost functional as given by (2.13) subject to (2.12). Recalling that the problem is now an entirely deterministic one, this is done in a most natural way with the aid of the variational calculus of operators, in particular, using the existence of the Gateaux differential. We will then on closing this section remark, without proof, that by the (assumed) existence of the minimizing pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\ \textrm{in}\ \mathcal{P}$$ the local minimum found is, in fact, the unique minimum. Let us turn briefly to the existence question. At this point, before entering into the discussion of necessary conditions, we would have to prove that the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$ actually exists in $$\mathcal{P}$$ or, at least, some (appropriate) subset thereof. Here we move directly to the question of necessary conditions, leaving the consideration of this particular synthesis problem to some future occasion. A brief outline of how such a proof might be carried out shall nevertheless be attempted. If one were to admit the possibility of a pair $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ being restrained to some subset of $$\mathcal{P}$$, one would have to begin by establishing weak-star compactness of bounded subsets of that space. The weak-star topology for $$\mathcal{P}$$ can be considered for this. In our case of unrestrained $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$, however, given the boundedness of any minimizing sequence of the pair in the space $$\mathcal{P}$$, the convergence of a subsequence in the weak-star topology is guaranteed in $$\mathcal{P}$$. Thus, using the immediately verified fact that the cost functional, as defined by (2.13), is positive, all that remains is to show that the map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ is lower semicontinuous on $$\mathcal{P}$$ in the weak-star topology. This by itself is an intricate matter, however, resulting in a rather lengthy proof (see, e.g. the paper by Ahmed, 2015), which we shall not go into here. For all the work in this section, we assume $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ attains a minimum on $$\mathcal{P}$$, i.e. there is a minimizing pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$. We begin with two lemmas which will be important later in the proof of our necessary conditions. Their proofs are elementary and are, therefore, omitted. For clarity and to avoid confusion in some parts of what follows, with some abuse of notation we sometimes write the covariance operator $${\mathscr{S}}\left (t\right )$$ as $${\mathscr{S}}\left (K\left (t\right ),L\left (t\right )\right )$$ to highlight its dependence on the feedback and estimator gain operators. To this end, we write $${\mathscr{S}}\left (K\left (t\right )\right )$$ or $${\mathscr{S}}\left (L\left (t\right )\right )$$ in place of $${\mathscr{S}}\left (K\left (t\right ),L\left (t\right )\right )$$, depending on whether we are to evaluate the Gateaux (partial) differential of the map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{\mathscr{S}}\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ at $$K\left (\;\!\cdot \;\!\right )$$ or $$L\left (\;\!\cdot \;\!\right )$$. In like vein, we also sometimes write $${J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$ or $${J}\left ({{L}}\left (\;\!\cdot \;\!\right )\right )$$ in place of $${J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$. In order to calculate the Gateaux differentials it will be necessary to introduce the operators $$F\left (t\right ):X\rightarrow \mathbb{R}^{m}$$ and $$G\left (t\right ):\mathbb{R}^{p}\rightarrow X$$ where $$F\left (t\right )\in \mathcal{L}\left (X,\mathbb{R}^{m}\right )$$ and $$G\left (t\right )\in \mathcal{L}\left (\mathbb{R}^{p},X\right )$$ for each t, and the maps $$t\mapsto F\left (t\right )$$ and $$t\mapsto G\left (t\right )$$ are both assumed to be strongly continuous on $$\left [t_{0},t_{1}\right ]$$. Set   $$ \mathscr{F}\left(t\right)=\left(\begin{array}{@{}cc@{}} F\left(t\right) & 0 \\ 0 & 0 \end{array}\right)\quad\textrm{and}\quad\mathscr{G}\left(t\right)=\left(\begin{array}{@{}cc@{}} 0 & G\left(t\right) \\ 0 & -G\left(t\right) \end{array}\right). $$ Lemma 3.1 Consider the system (2.12). The map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{\mathscr{S}}\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )$$ is Gateaux differentiable on $$\mathcal{P}$$, and the following assertions hold: (i) the Gateaux differential of $$K\left (\;\!\cdot \;\!\right )\mapsto{\mathscr{S}}\left (K\left (\;\!\cdot \;\!\right )\right )$$ is given by the solution for $$t_{0}\leqslant t\leqslant t_{1}$$ of   \begin{align*} \left\langle \dot{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{1},z_{2}\right\rangle &=\left\langle{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \left(\mathscr{A}\!+\!\mathscr{B}\left(t\right)\right)^{\ast} z_{1},{\mathscr{S}}^{\prime}\!\left(K\!\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \!+\!\left\langle{\mathscr{S}}\left(K\!\left(t\right)\right)z_{1},\mathscr{F}\left(t\right)^{\ast} \mathscr{B}^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \mathscr{F}\left(t\right)^{\ast} \mathscr{B}^{\ast} z_{1},{\mathscr{S}}\left(K\left(t\right)\right) z_{2}\right\rangle, \\{\mathscr{S}}^{\prime}\left(K\left(t_{0}\right);F\left(t_{0}\right)\right)&=0; \end{align*} (ii) the Gateaux differential of $$L\left (\;\!\cdot \;\!\right )\mapsto{\mathscr{S}}\left (L\left (\;\!\cdot \;\!\right )\right )$$ is given by the solution for $$t_{0}\leqslant t\leqslant t_{1}$$ of   \begin{align*} \left\langle \dot{\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)z_{1},z_{2}\right\rangle &=\left\langle{\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast} z_{1},{\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)z_{2}\right\rangle \!+\!\left\langle{\mathscr{S}}\left(L\left(t\right)\right)z_{1},\mathscr{C}^{\ast} \mathscr{G}\left(t\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \mathscr{C}^{\ast} \mathscr{G}\left(t\right)^{\ast} z_{1},{\mathscr{S}}\left(L\left(t\right)\right) z_{2}\right\rangle +\left\langle \mathscr{N}\mathscr{G}\left(t\right)^{\ast} z_{1},\mathscr{L}\left(t\right)^{\ast} z_{2}\right\rangle \\ &\quad+\left\langle \mathscr{L}\left(t\right)^{\ast} z_{1},\mathscr{N}\mathscr{G}\left(t\right)^{\ast} z_{2}\right\rangle, \\{\mathscr{S}}^{\prime}\left(L\left(t_{0}\right);G\left(t_{0}\right)\right)&=0. \end{align*} Lemma 3.2 Consider the cost functional (2.13). The map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ is Gateaux differentiable on $$\mathcal{P}$$, and the following assertions hold: (i) the Gateaux differential of $$K\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$ is given by   \begin{align*} {J}^{\prime}\!\left({{K}}\!\left(\;\!\cdot\;\!\right)\!;{{F\!}}\left(\;\!\cdot\;\!\right)\right)\!&=\!\frac{1}{2}\operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\!\left(K\!\left(t_{1}\right)\!;F\!\left(t_{1}\right)\right)\\ &\quad+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left[\left(\mathscr{Q}\!+\!\mathscr{K}\!\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\!\left(K\!\left(t\right);F\left(t\right)\right)\right.\\ &\quad\left.+\left(\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{F}\left(t\right)+\mathscr{F}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}\left(K\left(t\right)\right)\right]\,\mathrm{d}t; \end{align*} (ii) the Gateaux differential of $$L\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{L}}\left (\;\!\cdot \;\!\right )\right )$$ is given by   \begin{align*} {J}^{\prime}\left({{L}}\left(\;\!\cdot\;\!\right);{{G}}\left(\;\!\cdot\;\!\right)\right)&=\frac{1}{2}\operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\left(L\left(t_{1}\right);G\left(t_{1}\right)\right)\\[-2pt]&\quad+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\left(L\left(t\right);G\left(t\right)\right)\,\mathrm{d}t. \end{align*} The results in Lemma 3.2, as they are at the moment, are difficult to evaluate and do not help us establish necessary conditions in a form that we shall find useful. The source of the difficulty is the appearance of $${\mathscr{S}}^{\prime }\left (K\left (\;\!\cdot \;\!\right );F\left (\;\!\cdot \;\!\right )\right )$$ and $${\mathscr{S}}^{\prime }\left (L\left (\;\!\cdot \;\!\right );G\left (\;\!\cdot \;\!\right )\right )$$, respectively, in the expressions for $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$ and $${J}^{\prime }\left ({{L}}\left (\;\!\cdot \;\!\right );{{G}}\left (\;\!\cdot \;\!\right )\right )$$. Still, they will play a crucial role in the following, and we shall see presently that a very simple manipulation provides a way round this problem. Before we do this, however, we make use of a standard result to state the following lemma, thereby introducing the so-called ‘adjoint’ system. Lemma 3.3 For $$z_{1},z_{2}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ the function $$\left\langle{\mathscr{P}}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ exists and is the unique selfadjoint solution of the system   \begin{align} \left\langle \dot{\mathscr{P}}\left(t\right)z_{1},z_{2}\right\rangle &=-\left\langle{\mathscr{P}}\left(t\right)z_{1},\left(\mathscr{A}+\mathscr{B}\left(t\right)\right) z_{2}\right\rangle -\left\langle \left(\mathscr{A}+\mathscr{B}\left(t\right)\right) z_{1},{\mathscr{P}}\left(t\right)z_{2}\right\rangle \nonumber\\ &\quad-\left\langle \mathscr{Q}z_{1},z_{2}\right\rangle -\left\langle \mathscr{R}\mathscr{K}\left(t\right)z_{1},\mathscr{K}\left(t\right)z_{2}\right\rangle ,\quad \quad t_{0}< t< t_{1},\\\nonumber\quad{\mathscr{P}}\left(t_{1}\right)&=\mathscr{Q}. \end{align} (3.1)Furthermore, $${\mathscr{P}}\left (t\right )\in \mathcal{L}\left (Z\right )$$, which is positive and nuclear for each fixed value of t, and $${\mathscr{P}}\left (\;\!\cdot \;\!\right )z_{1}\in \boldsymbol{C}\left (\left [t_{0},t_{1}\right ];Z\right )$$. Proof We sketch the proof. The integrated form of (3.1) is, for any $$z_{1}\in Z$$,   $$ \mathscr{P}\left(t\right)z_{1}=\mathscr{U}\left(t_{1},t\right)^{\ast}\mathscr{Q}\mathscr{U}\left(t_{1},t\right) z_{1}+\int^{t_{1}}_{t}\mathscr{U}\left({s},t\right)^{\ast}\left(\mathscr{Q}+\mathscr{K}\left({s}\right)^{\ast}\mathscr{R}\mathscr{K}\left({s}\right)\right)\mathscr{U}\left({s},t\right) z_{1}\,\,\mathrm{d}{s}. $$We know that $$\mathscr{Q}\in \mathcal{L}\left (Z\right )$$ and is positive symmetric. In the preceding section we have assumed implicitly that $$K\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (X,\mathbb{R}^{m}\right )\right )$$ and $$L\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (\mathbb{R}^{p},X\right )\right )$$, whence $$\mathscr{K}\left (t\right )^{\ast }\mathscr{R}\mathscr{K}\left (t\right )\in \mathcal{L}\left (Z\right )$$. Thus, $${\mathscr{P}}\left (t\right )\in \mathcal{L}\left (Z\right )$$ and is, for each t, positive symmetric, and hence selfadjoint. In fact, in keeping with the remarks following (2.11), we see immediately that $${\mathscr{P}}\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (Z\right )\right )$$. Further, it follows from the nuclearity of $$\mathscr{Q}$$ that $${\mathscr{P}}\left (t\right )$$ is nuclear for $$t_{0}\leqslant t\leqslant t_{1}$$. The existence of a unique strongly continuous solution of (3.1) can be shown as follows. We first note that we need not assume here (as we do in the case of (2.11)) that $$\mathscr{A}+\mathscr{B}\left (\;\!\cdot \;\!\right )$$ is the generator of a strong evolution family on Z, because the function $$t\mapsto \left\langle{\mathscr{P}}\left (t\right )z_{1},z_{2}\right\rangle $$ must by definition be differentiable on $$\left (t_{0},t_{1}\right )$$ to give (3.1). This requires, however, that we proceed somewhat indirectly when it comes to proving uniqueness, by exploiting the density of $$\boldsymbol{D}\left (\mathscr{A}\right )$$ in Z. Let us suppose that $$\mathscr{B}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );\mathcal{L}\left (Z\right )\right )$$, so that the evolution family is strong. We may suppose further for the moment that $$\mathscr{D}\left (\;\!\cdot \;\!\right ){{r}}\left (\;\!\cdot \;\!\right )\in \boldsymbol{C}^{1}\left (\left (t_{0},t_{1}\right );Z\right )$$. Then for $$z_{0}\in \boldsymbol{D}\left (\mathscr{A}\right )$$, we may differentiate the function $$t\mapsto \left\langle \left ({\mathscr{P}}\left (t\right )-{\mathscr{M}}\left (t\right )\right ){{z}}\left (t\right ),{{z}}\left (t\right )\right\rangle $$, where $$\left\langle{\mathscr{M}}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ for $$z_{1},z_{2}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ is another solution of (3.1), to get $${\mathscr{P}}\left (\;\!\cdot \;\!\right )={\mathscr{M}}\left (\;\!\cdot \;\!\right )$$. The results so far provide the required ingredients for the following development of necessary conditions. Proposition 3.1 Necessary conditions for a pair $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ to extremize $${J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ as defined by (2.13) are that, for $$t_{0}\leqslant t\leqslant t_{1}$$,   \begin{align} \operatorname{Tr}\mathscr{F}\!\left(t\right)^{\ast}\left(\mathscr{B}^{\ast}{\mathscr{P}}\!\left(t\right)+\mathscr{R}\mathscr{K}\!\left(t\right)\right){\mathscr{S}}\!\left(t\right)\!=\!0\quad\textrm{and}\quad\operatorname{Tr}{\mathscr{P}}\!\left(t\right)\left({\mathscr{S}}\!\left(t\right)\mathscr{C}^{\ast}+\mathscr{L}\!\left(t\right)\mathscr{N}\right)\mathscr{G}\left(t\right)^{\ast}\!=\!0, \end{align} (3.2)where for $$z_{1},z_{2}$$ in $$\boldsymbol{D}\left (\mathscr{A}\right )$$ or $$\boldsymbol{D}\left (\mathscr{A}^{\ast }\right )$$, as appropriate, the functions $$\left\langle \mathscr{P}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ and $$\left\langle \mathscr{S}\left (\;\!\cdot \;\!\right )z_{1},z_{2}\right\rangle $$ satisfy (3.1) and (2.12), respectively. Proof We only prove the first necessary condition in (3.2) as the other is proven in a similar manner. Recall from Lemma 3.2(i) the expression for $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$. To begin with, we first of all show how to eliminate $${\mathscr{S}}^{\prime }\left (K\left (\;\!\cdot \;\!\right );F\left (\;\!\cdot \;\!\right )\right )$$ there, and in order to do so we compute formally, by using the adjoint system in Lemma 3.3, that   \begin{align*} \frac{{d}}{{d}t}&\left\langle{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)z_{1},z_{2}\right\rangle\\ &=\frac{{d}}{{d}t}\left\langle{\mathscr{P}}\left(t\right)z_{1},{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \\ &=-\left\langle \left[\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast}{\mathscr{P}}\left(t\right)+{\mathscr{P}}\left(t\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)+\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right]z_{1},{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \\ &\quad+\left\langle{\mathscr{P}}\left(t\right)z_{1},\dot{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)z_{2}\right\rangle \\ &=-\left\langle \left[{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast}{\mathscr{P}}\left(t\right)+{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)\right.\right. \\ &\qquad\quad-\dot{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)\!\!\left.\left.\vphantom{\big[}\right]z_{1},z_{2}\right\rangle -\left\langle{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right)z_{1},z_{2}\right\rangle \end{align*}for $$z_{1}\in \boldsymbol{D}\left (\mathscr{A}\right )$$ and $$z_{2}\in \boldsymbol{D}\left (\mathscr{A}^{\ast }\right )$$. Then since all terms are integrable,   \begin{align*} &\left\langle{\mathscr{S}}^{\prime}\left(K\left(t_{1}\right);F\left(t_{1}\right)\right)\mathscr{Q}z_{1},z_{2}\right\rangle \\ &\quad=-\int^{t_{1}}_{t_{0}}\left\langle \left[{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)^{\ast}{\mathscr{P}}\left(t\right)+{\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right){\mathscr{P}}\left(t\right)\left(\mathscr{A}+\mathscr{B}\left(t\right)\right)\right. \right. \\ &\qquad\qquad\qquad-\dot{\mathscr{S}}^{\prime}\!\left(K\!\left(t\right)\!;F\left(t\right)\right)\!{\mathscr{P}}\!\left(t\right)\!\!\left.\left.\vphantom{\big[}\right]z_{1},z_{2}\right\rangle \,\mathrm{\!d}t -\!\int^{t_{1}}_{t_{0}}\!\left\langle{\mathscr{S}}^{\prime}\!\left(K\!\left(t\right);F\!\left(t\right)\right)\left(\mathscr{Q}\!+\!\mathscr{K}\!\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\!\left(t\right)\right)z_{1},z_{2}\right\rangle \,\mathrm{d}t, \end{align*}where we have taken into account the initial and end conditions (see Lemma 3.1(i) and Lemma 3.3)   $$ {\mathscr{S}}^{\prime}\left(K\left(t_{0}\right);F\left(t_{0}\right)\right)=0,\quad{\mathscr{P}}\left(t_{1}\right)=\mathscr{Q}. $$Taking the trace of both sides, we obtain in view of Lemma 3.1(i), and using the cyclic properties of the trace operation,   \begin{align*} \operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\left(K\left(t_{1}\right);F\left(t_{1}\right)\right)&=\int^{t_{1}}_{t_{0}}\operatorname{Tr}{\mathscr{P}}\left(t\right)\left(\mathscr{B}\mathscr{F}\left(t\right){\mathscr{S}}\left(K\left(t\right)\right)+{\mathscr{S}}\left(K\left(t\right)\right)\mathscr{F}\left(t\right)^{\ast}\mathscr{B}^{\ast}\right)\,\mathrm{d}t \\ &\quad-\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\,\mathrm{d}t. \end{align*}Thus, what we have shown is that   \begin{align*} \operatorname{Tr}{\mathscr{Q}}{\mathscr{S}}^{\prime}\left(K\left(t_{1}\right);F\left(t_{1}\right)\right)=&\int^{t_{1}}_{t_{0}}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Tr}{\mathscr{P}}\left(t\right){\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\,\mathrm{d}t \\ =&\int^{t_{1}}_{t_{0}}\operatorname{Tr}{\mathscr{P}}\left(t\right)\left(\mathscr{B}\mathscr{F}\left(t\right){\mathscr{S}}\left(K\left(t\right)\right)+{\mathscr{S}}\left(K\left(t\right)\right)\mathscr{F}\left(t\right)^{\ast}\mathscr{B}^{\ast}\right)\,\mathrm{d}t \\ &-\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}^{\prime}\left(K\left(t\right);F\left(t\right)\right)\,\mathrm{d}t. \end{align*}Substituting the above relation into the original expression for $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$ now leads to   \begin{align*} {J}^{\prime}\left({{K}}\left(\;\!\cdot\;\!\right);{{F}}\left(\;\!\cdot\;\!\right)\right)&=\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}{\mathscr{P}}\left(t\right)\left(\mathscr{B}\mathscr{F}\left(t\right){\mathscr{S}}\left(K\left(t\right)\right)+{\mathscr{S}}\left(K\left(t\right)\right)\mathscr{F}\left(t\right)^{\ast}\mathscr{B}^{\ast}\right)\,\mathrm{d}t \\[-2pt] &\quad+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}\left(\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{F}\left(t\right)+\mathscr{F}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}\left(K\left(t\right)\right)\,\mathrm{d}t. \end{align*}All terms containing $${\mathscr{S}}^{\prime }\left (K\left (\;\!\cdot \;\!\right );F\left (\;\!\cdot \;\!\right )\right )$$ have been eliminated from the original $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )$$ at this point, and it immediately follows, on reordering the traces, that   $$ {J}^{\prime}\left({{K}}\left(\;\!\cdot\;\!\right);{{F}}\left(\;\!\cdot\;\!\right)\right)=\int^{t_{1}}_{t_{0}}\operatorname{Tr}\mathscr{F}\left(t\right)^{\ast}\left(\mathscr{B}^{\ast}{\mathscr{P}}\left(t\right)+\mathscr{R}\mathscr{K}\left(t\right)\right){\mathscr{S}}\left(K\left(t\right)\right)\,\mathrm{d}t. $$In order that $$K\left (\;\!\cdot \;\!\right )\in{\boldsymbol{P}}_{\infty }\left (\left [t_{0},t_{1}\right ];\mathcal{L}\left (X,\mathbb{R}^{m}\right )\right )$$ furnish an extremum of the map $$K\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$, it is necessary that $${J}^{\prime }\left ({{K}}\left (\;\!\cdot \;\!\right );{{F}}\left (\;\!\cdot \;\!\right )\right )=0$$, and we clearly have the first necessary condition in (3.2), completing the first part of the proposition. The result of Proposition 3.1 enables us to obtain—without using Hilbert space probability theory—an alternative generalization of the separation principle to encompass infinite-dimensional systems. This is our main result, which is embodied in the next theorem and its corollary; but before we go into the statement and proof of the theorem, it will be necessary to decompose the operators $${\mathscr{P}}\left (t\right )\in \mathcal{L}\left (Z\right )$$ and $${\mathscr{S}}\left (t\right )\in \mathcal{L}\left (Z\right )$$ according to the decompositions   $$ {\mathscr{P}}\left(t\right)=\left(\begin{array}{@{}cc@{}} P_{11}\left(t\right) & P_{12}\left(t\right) \\ P_{21}\left(t\right) & P_{22}\left(t\right) \end{array}\right)\quad\textrm{and}\quad{\mathscr{S}}\left(t\right)=\left(\begin{array}{@{}cc@{}} S_{11}\left(t\right) & S_{12}\left(t\right) \\ S_{21}\left(t\right) & S_{22}\left(t\right) \end{array}\right). $$Since $${\mathscr{P}}\left (t\right )$$ and $${\mathscr{S}}\left (t\right )$$ are both symmetric (which follows from our remarks following the statement of (2.11), and Lemma 3.3), it holds that   $$ P_{21}\left(t\right)=P_{12}\left(t\right)^{\ast}\quad\textrm{and}\quad S_{21}\left(t\right)=S_{12}\left(t\right)^{\ast}. $$ Theorem 3.1 Consider the system (2.12), and let the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ minimize the cost functional defined by (2.13). Then we have, for $$t_{0}\leqslant t\leqslant t_{1}$$, that   \begin{align} K^{o}\left(t\right)=-R^{-1}B^{\ast} P_{11}\left(t\right)\quad\textrm{and}\quad L^{o}\left(t\right)=S_{22}\left(t\right)C^{\ast} W^{-1}, \end{align} (3.3)where for $$x_{1},x_{2}$$ in $$\boldsymbol{D}\left ({A}\right )$$ or $$\boldsymbol{D}\left ({A}^{\ast }\right )$$, as appropriate, the functions $$\left\langle P_{11}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ and $$\left\langle S_{22}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ exist and are, respectively, the unique selfadjoint solutions for $$t_{0}\leqslant t\leqslant t_{1}$$ of   \begin{align*} \left\langle \dot{P}_{11}\left(t\right)x_{1},x_{2}\right\rangle &=-\big\langle{P}_{11}\left(t\right)x_{1},{A} x_{2}\big\rangle -\big\langle{A} x_{1},{P}_{11}\left(t\right)x_{2}\big\rangle -\big\langle Qx_{1},x_{2}\big\rangle +\big\langle{R}^{-1}B^{\ast}{P}_{11}\left(t\right) x_{1},B^{\ast} {P}_{11}\left(t\right) x_{2}\big\rangle ,\\[-2pt] {P}_{11}\left(t_{1}\right)&=Q, \end{align*}and   \begin{align*} \left\langle \dot{S}_{22}\left(t\right)x_{1},x_{2}\right\rangle &=\left\langle{S}_{22}\left(t\right)x_{1},{A}^{\ast} x_{2}\right\rangle +\left\langle{A}^{\ast} x_{1},{S}_{22}\left(t\right)x_{2}\right\rangle +\left\langle VD^{\ast} x_{1},D^{\ast} x_{2}\right\rangle -\big\langle W^{-1} C{S}_{22}\left(t\right) x_{1},C{S}_{22}\left(t\right)x_{2}\big\rangle , \\[-2pt]{S}_{22}\left(t_{0}\right)&=S_{0}. \end{align*} Remark 3.1 As we shall see in the proof of the theorem (and as is already apparent), the differential operator equations in the theorem are the inner product versions of the well-known Riccati equations of the finite-dimensional LQG theory. These equations have been studied extensively (the earliest studies, by Kalman, 1960 and Wonham, 1968a, go back to the beginnings of optimal control theory in the 1960s), and many properties such as existence, uniqueness, continuity and asymptotic behaviour of their solutions are known. For the generalization of the finite dimensional results to the infinite dimensional case of interest here, see the books by Curtain & Pritchard (1978) and Balakrishnan (1981) or, to some extent, the book by Curtain & Zwart (1995). Proof of Theorem 3.1 We deduced in Proposition 3.1 that in seeking the extrema of the maps $$K\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right )\right )$$ and $$L\left (\;\!\cdot \;\!\right )\mapsto{J}\left ({{L}}\left (\;\!\cdot \;\!\right )\right )$$ simultaneously over $$\mathcal{P}$$ it is necessary that the conditions in (3.2) hold. Using the partitioned elements for $${\mathscr{P}}\left (t\right )$$ and $${\mathscr{S}}\left (t\right )$$ in (3.2), it may be seen that, since the pair $$\left (F\left (\;\!\cdot \;\!\right ),G\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ is arbitrary, this will be achieved if and only if for $$t_{0}\leqslant t\leqslant t_{1}$$ we have that   \begin{align} (B^{\ast}{P}_{11}(t)+RK(t)){S}_{11}(t)+B^{\ast}{P}_{12}(t){S}_{12}(t)^{\ast}=0 \end{align} (3.4)and   \begin{align} ({P}_{11}(t)-{P}_{12}(t)^{\ast})({S}_{12}(t)C^{\ast} +L(t) W)+({P}_{12}(t)-{P}_{22}(t))({S}_{22}(t)C^{\ast}-L(t) W)=0. \end{align} (3.5)We find that making use of the partitioned expressions for $${\mathscr{P}}\left (t\right )$$ and $${\mathscr{S}}\left (t\right )$$ in (3.1) and (2.12) will yield the decomposition into, in each case, four coupled differential operator equations. In view then of the fact (this can be verified by routine computations) that   \begin{align} P_{11}\left(\;\!\cdot\;\!\right)=P_{12}\left(\;\!\cdot\;\!\right)^{\ast}\quad\textrm{and}\quad S_{12}\left(\;\!\cdot\;\!\right)=0, \end{align} (3.6)we obtain from the two relations (3.4) and (3.5) the expressions for the gain operators in (3.3) together with the resulting problems in the statement of the theorem for the Riccati equations (which are simply the appropriate partitionings of (3.1) and (2.12) with the expressions in (3.3) substituted in them). The existence and uniqueness of the functions $$\left\langle P_{11}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ and $$\left\langle S_{22}\left (\;\!\cdot \;\!\right )x_{1},x_{2}\right\rangle $$ are standard results (Balakrishnan, 1981, Sections 5.2 and 6.8); refer also to the proof of Lemma 3.3. The fact that $${P}_{11}\left (t\right )$$ and $${S}_{22}\left (t\right )$$ are selfadjoint is obvious. What remains to be verified is that the joint extremum found is, in fact, a minimum. To begin with, we note, first of all, that the quantities $$\operatorname{Tr}\mathscr{Q}\mathscr{S}\left (t_{1}\right )$$ and $$\operatorname{Tr}\left (\mathscr{Q}+\mathscr{K}\left (t\right )^{\ast }\mathscr{R}\mathscr{K}\left (t\right )\right )\mathscr{S}\left (t\right )$$ which appear in the cost functional (2.13) can be partitioned as   $$ \operatorname{Tr}\mathscr{Q}\mathscr{S}\left(t_{1}\right)=\operatorname{Tr}QS_{11}\left(t_{1}\right)+\operatorname{Tr}QS_{22}\left(t_{1}\right) $$and   $$ \operatorname{Tr}\left(\mathscr{Q}+\mathscr{K}\left(t\right)^{\ast}\mathscr{R}\mathscr{K}\left(t\right)\right)\mathscr{S}\left(t\right)=\operatorname{Tr}\left(Q+ K\left(t\right)^{\ast} RK\left(t\right)\right)S_{11}\left(t\right)+\operatorname{Tr}QS_{22}\left(t\right), $$where, in both, we have taken account of the latter condition in (3.6); and so,   \begin{align} {J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right)&=\frac{1}{2}\operatorname{Tr}QS_{11}\left(t_{1}\right)+\frac{1}{2}\operatorname{Tr}QS_{22}\left(t_{1}\right)\nonumber\\[-2pt] &\quad+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}\left(Q+K\left(t\right)^{\ast} RK\left(t\right)\right)S_{11}\left(t\right)\,\mathrm{d}t +\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}QS_{22}\left(t\right)\,\mathrm{d}t. \end{align} (3.7)It follows then from calculations similar to those made in the proof of Proposition 3.1, and using the corresponding Riccati equations in the statement of the theorem, that   \begin{align*} \operatorname{Tr}QS_{11}\left(t_{1}\right)&=\int^{t_{1}}_{t_{0}}\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{Tr}P_{11}\left(t\right)S_{11}\left(t\right)\,\mathrm{d}t \\[-2pt] &=\int^{t_{1}}_{t_{0}}\operatorname{Tr}{P}_{11}\left(t\right)S_{22}\left(t\right)C^{\ast} W^{-1}CS_{22}\left(t\right)\,\mathrm{d}t\!-\!\int^{t_{1}}_{t_{0}}\operatorname{Tr}\big(Q+ P_{11}\left(t\right)BR^{-1}B^{\ast} P_{11}\left(t\right)\big)S_{11}\left(t\right)\,\mathrm{d}t. \end{align*}With this, and rearranging, the right-hand side of (3.7) becomes   \begin{multline*} \frac{1}{2}\operatorname{Tr}QS_{22}\left(t_{1}\right)+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}QS_{22}\left(t\right)\,\mathrm{d}t+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}{P}_{11}\left(t\right)S_{22}\left(t\right)C^{\ast} W^{-1}CS_{22}\left(t\right)\,\mathrm{d}t \\[-2pt] +\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}\big(K\left(t\right)^{\ast} RK\left(t\right)-P_{11}\left(t\right)BR^{-1}B^{\ast} P_{11}\left(t\right)\big)S_{11}\left(t\right)\,\mathrm{d}t, \end{multline*}and it is readily seen that $${J}\left ({{K}}^{o}\left (\;\!\cdot \;\!\right ),{{L}}^{o}\left (\;\!\cdot \;\!\right )\right )\leqslant{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$. This completes the proof. If we had proven the existence of the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$, we would have shown that the map $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\mapsto{J}\left ({{K}}\left (\;\!\cdot \;\!\right ),{{L}}\left (\;\!\cdot \;\!\right )\right )$$ is lower semicontinuous (in the weak-star sense) on $$\mathcal{P}$$. Assuming this result, as we did in the theorem, the fact that then the pair is unique lends to the following strengthening of Theorem 3.1. Corollary 3.1 The minimizing pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$ is unique and is given by the individual expressions in (3.3). Then   \begin{align} {J}\left({{K}}^{o}\left(\;\!\cdot\;\!\right),{{L}}^{o}\left(\;\!\cdot\;\!\right)\right)\leqslant{J}\left({{K}}\left(\;\!\cdot\;\!\right),{{L}}\left(\;\!\cdot\;\!\right)\right) \end{align} (3.8)for all $$\left (K\left (\;\!\cdot \;\!\right ),L\left (\;\!\cdot \;\!\right )\right )\in \mathcal{P}$$, and the minimal cost, realized with the pair $$\left (K^{o}\left (\;\!\cdot \;\!\right ),L^{o}\left (\;\!\cdot \;\!\right )\right )$$, is given by   $$ {J}\left({{K}}^{o}\left(\;\!\cdot\;\!\right),{{L}}^{o}\left(\;\!\cdot\;\!\right)\right)=\frac{1}{2}\operatorname{Tr}QS_{22}\left(t_{1}\right)+\frac{1}{2}\int_{t_{0}}^{t_{1}}\operatorname{Tr}QS_{22}\left(t\right)\,\mathrm{d}t+\frac{1}{2}\int^{t_{1}}_{t_{0}}\operatorname{Tr}{P}_{11}\left(t\right)S_{22}\left(t\right)C^{\ast} W^{-1}CS_{22}\left(t\right)\,\mathrm{d}t. $$ 4. Concluding remarks In this paper an approach different from the common probabilistic ones has been considered to derive a solution to the infinite-dimensional LQG regulator problem. By transforming the problem into a deterministic framework we were able to derive, using standard variational arguments, a set of necessary conditions for the cost functional involved to take on an extremum (in fact, a unique minimum). In particular, we obtained a new method of proof for the separation principle that, because we consider the minimization problem on an extended Hilbert state-space, completely avoids abstract probability theory. References Ahmed, N. U. & Li, P. ( 1991) Quadratic regulator theory and linear filtering under system constraints. IMA J. Math. Control Inf. , 8, 93-- 107. Google Scholar CrossRef Search ADS   Ahmed, N. U. ( 1998) Linear and Nonlinear Filtering for Scientists and Engineers . Singapore: World Scientific. Google Scholar CrossRef Search ADS   Ahmed, N. U. ( 2015) Infinite dimensional uncertain dynamic systems on Banach spaces and their optimal output feedback control. Discuss. Math. Differ. Incl. Control Optim ., 35, 65-- 87. Google Scholar CrossRef Search ADS   Balakrishnan, A. V. ( 1981) Applied Functional Analysis . New York: Springer. Curtain, R. F. & Pritchard, A. J. ( 1978) Infinite Dimensional Linear Systems Theory . Berlin: Springer. Google Scholar CrossRef Search ADS   Curtain, R. F. & Zwart, H. J. ( 1995) An Introduction to Infinite-Dimensional Linear Systems Theory . New York: Springer. Google Scholar CrossRef Search ADS   Kalman, R. E. ( 1960) Contributions to the theory of optimal control. Bol. Soc. Mat. Mex. , 5, 102-- 119. Phillips, R. S. ( 1953) Perturbation theory for semi-groups of linear operators. Trans. Am. Math. Soc. , 74, 199-- 221. Google Scholar CrossRef Search ADS   Wonham, W. M. ( 1968a) On a matrix Riccati equation of stochastic control. SIAM J. Control , 6, 681-- 697. Google Scholar CrossRef Search ADS   Wonham, W. M. ( 1968b) On the separation theorem of stochastic control. SIAM J. Control , 6, 312-- 326. Google Scholar CrossRef Search ADS   © 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

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Mar 21, 2018

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