TY - JOUR
AU - Bivziuk, V O
AB - Abstract The paper deals with the problem of stabilizing the equilibrium states of a family of non-linear non-autonomous systems. It is assumed that the nominal system is a linear controlled system with periodic coefficients. For the nominal controlled system, a new method for constructing a Lyapunov function in the quadratic form with a variable matrix is proposed. This matrix is defined as an approximate solution of the Lyapunov matrix differential equation in the form of a piecewise exponential function based on partial sums of a W. Magnus series. A stabilizing control in the form of a linear feedback with a piecewise constant periodic matrix is constructed. This control simultaneously stabilizes the considered family of systems. The estimates of the domain of attraction of an asymptotically stable equilibrium state of a closed-loop system that are common for all systems are obtained. A numerical example is given. 1. Introduction Stabilization of non-linear controlled systems with functional uncertainties is an important problem of modern control theory. Robust and adaptive stabilization of non-linear systems with uncertain input is considered in Zak (2002), Amato (2006) and Zuber & Gelig (2008). A significant number of papers are devoted to the stability study for systems with functional uncertainties, in particular Takagi–Sugeno fuzzy models, which also belong to this class of non-linear systems (Yu et al., 2005; Jabri et al., 2012; Slyn’ko & Denysenko, 2014). The problem of constructing a stabilizing control for a family of non-linear controlled systems with functional uncertainties is studied in Nguang & Fu (1998) and Korobov & Lutsenko (2014), where the nominal system is an autonomous system. In Nguang & Fu (1998), the global quadratic stabilization problem for a certain class of non-linear systems is considered. In this paper, using the well-known Riccati approach, a feedback control design method that quadratically stabilizes the system for all permissible uncertainties has been developed. This control law consists of linear and non-linear control elements. The local stabilization problem and estimation of the general domain of attraction for a family of non-linear controlled systems with a nominal autonomous affine system are studied in Korobov & Lutsenko (2014). An important result of this work is an explicit formula for parameters of the feedback function that stabilizes all the systems of the considered family. In Polyak & Scherbakov (2002), a simultaneous stabilization approach based on quadratic stabilization methods is used, which ensures the existence of a common Lyapunov function for a given family of systems. In Polyak & Scherbakov (2002), the estimates of the common domain of attraction, which are invariant under uncertainties, are obtained. Firstly, the aim of this paper is to generalize the results of Korobov & Lutsenko (2014) for the case of affine-periodic nominal system. It is worth mentioning that periodic systems are very important for modelling systems with variable parameters and adaptive controlled systems. For example, these systems are typical models of satellites or helicopter rotors. It is well known that the phenomenon of parametric resonance can have negative influence on the system behaviour; therefore, it is important to develop methods for synthesizing controls that stabilize resonance oscillations. Note that the problem of stabilizing linear periodic systems is well developed only for linear discrete-time systems (De Souza & Trofino, 2000; Nguyen et al., 2017). In De Souza & Trofino (2000), the stabilization problem is considered for linear discrete-time periodic systems. Based on the approach of linear matrix inequalities (LMIs), the stabilizability conditions are obtained through static periodic state feedback and also through static periodic output feedback signals. For linear periodic systems, this problem is sufficiently developed in the case when the coefficients of the system are piecewise constant functions. For example, stability conditions are obtained in the recent paper (Li et al., 2018), which considers the stabilization problem for linear periodic systems with piecewise constant coefficients using polynomial approximations of solutions of the Lyapunov matrix differential equation. Secondly, we propose an approach for synthesizing controls in the form of linear feedback with a piecewise constant time periodic matrix. In other words, the corresponding closed-loop system is a switched system (Liberzon & Morse, 1999). In order to obtain the stabilizability conditions, we use the discretization method, which has gained considerable popularity due to the significant progress of computational methods and means. Among the papers devoted to the discretization method in the problem of constructing Lyapunov functions, we mention Allerhand & Shaked (2011), Xiang & Xiao (2014) and Chen et al. (2017). In Xiang & Xiao (2014), the discretization method is used to study the stability of linear switched systems, all subsystems of which are unstable. In Chen et al. (2017), this method is used to construct the Lyapunov–Krasovskii functional and to study the stability of linear impulsive systems with delay. Using the discretization method, the problem of stability for linear switched systems has been investigated in Allerhand & Shaked (2011). Note that in these papers the discretization method is based on piecewise linear approximations of solutions of Lyapunov matrix differential equations. The contribution of this paper is multifarious. In fact, we first propose a new approach for the construction of a quadratic Lyapunov function with a time-dependent matrix. This approach is based on the discretization method. Unlike the classical version of the discretization method (Allerhand & Shaked, 2011; Xiang & Xiao, 2014; Chen et al., 2017), where piecewise linear approximations of the solution of the Lyapunov matrix differential equation are applied, we introduce piecewise exponential approximations. The application of functions of this class leads necessity to use the formalism of commutator calculus, in particular the Hausdorff identities. This circumstance closely relates our method to the Lie algebraic methods, which were previously used in the theory of the stability of motion in, e.g., Liberzon et al. (1999), Morin et al. (1999), Agrachev et al. (2012), Zuyev (2016), Grushkovskaya & Zuyev (2018), Slyn’ko (2018) and Slyn’ko & Tunç (2019). Then, we use the comparison method, with the comparison equation being a differential equation with impulse action. Similar comparison equations are used in Chatterjee & Liberzon (2006) to study the stability in terms of two measures of switched systems. After that, we propose a feedback control for a family of non-linear control systems to guarantee the asymptotic stability of a closed-loop system and obtain an estimate of the domain of attraction common to all systems. The matrix of the proposed linear feedback is piecewise constant. This result provides a significant contribution compared to the known results that can be found in the literature, since, for example, in the recent paper (Li et al., 2018) the stabilization problem has been solved with piecewise constant coefficients for a linear periodic system. Finally, we note that the introduced stability and stabilizability conditions essentially take into account the commutation properties of the matrices of the right-hand side of the nominal system. The paper is organized as follows. In Section 2, we present auxiliary algebraic results and stability conditions for a non-linear impulsive differential comparison equation. The robust stabilization problem of a family of non-linear control systems is formulated in Section 3. Section 4 is devoted to the construction of a Lyapunov function and its estimates. The main result of the paper is formulated in Section 5. Section 6 gives an example, and Section 7 is devoted to the discussion of the results. 2. Auxiliary results In this section, we remind some basic algebraic concepts (see, e.g., Magnus, 1954) needed to obtain the main results of this paper. The commutator of two matrices |$A, B\in \mathbb{R}^{n\times n}$| is defined by the formula $$\begin{equation*} [A,B]=AB-BA \end{equation*}$$ and introduces the Lie algebra structure in the associative matrix algebra |$\mathbb{R}^{n\times n}$|⁠. The commutation operator |$\operatorname{ad\,}_A$| (⁠|$A\in \mathbb{R}^{n\times n}$|⁠) is defined as a linear mapping from |$\mathbb{R}^{n\times n}$| to |$\mathbb{R}^{n\times n}$|⁠: $$\begin{equation*} \quad \mathbb{R}^{n\times n}\ni Y\mapsto\operatorname{ad\,}_A(Y)=[A,Y]. \end{equation*}$$ Let |$F(X,Y)$| be a formal series of matrix variables |$ X $| and |$ Y $|⁠, |$Z$| be a matrix variable, and |$\lambda \in \mathbb{R}$|⁠. Then the polarization identity $$\begin{equation*} F(X+\lambda Z,Y)=F(X,Y)+\lambda F_1(X,Y,Z)+\lambda^2F_2(X,Y,Z)+\dots \end{equation*}$$ determines the Hausdorff derivative |$\Big (Z\frac{\partial } {\partial X}\Big )F(X,Y)\overset{def}=F_1(X,Y,Z)$|⁠. Next, we define recursively the Lie polynomials in variables |$X$|⁠, |$Y$|⁠. (More detailed information about the Lie elements can be found in Magnus, 1954.) We have $$\begin{equation*} \{Y,X^0\}=Y,\quad \{Y,X^{l+1}\}=[\{Y,X^l\},X],\quad l\in\mathbb N. \end{equation*}$$ The element |$ e ^ X $| is defined by the following formula: $$\begin{equation*} e^X=\sum\limits_{k=0}^{\infty}\frac{1}{k!}X^k. \end{equation*}$$ For matrix variables |$X$| and |$Y$|⁠, the following Hausdorff identities are valid: $$\begin{equation} \begin{gathered} e^{-X}\Bigg(\Bigg(Y\frac{\partial}{\partial X}\Bigg) e^X\Bigg)=\Bigg\{Y,\frac{e^X-1}{X}\Bigg\}=Y+\sum\limits_{k=1}^{\infty}\frac{1}{(k+1)!}\{Y,X^k\},\\ \Bigg(\Bigg(Y\frac{\partial}{\partial X}\Bigg) e^X\Bigg)e^{-X}=\Bigg\{Y,\frac{1-e^{-X}}{X}\Bigg\}=Y+\sum\limits_{k=1}^{\infty}\frac{(-1)^k}{(k+1)!}\{Y,X^k\}. \end{gathered} \end{equation}$$(2.1) Let |$\mathbb{R}^{n\times n}$| be a finite-dimensional Banach algebra of square matrices of order |$ n $| with the spectral norm: |$\|A\|=\lambda _{\max }^{1/2}(A^{\operatorname{T\,}}A)$|⁠, |$A\in \mathbb{R}^{n\times n}$|⁠, where |$\lambda _{\max }(\cdot )$| is the maximal eigenvalue of the corresponding symmetric matrix. For two symmetric matrices |$P,\,Q\in \mathbb{R}^{n\times n}$|⁠, the notation |$P\prec Q$| means that the matrix |$Q-P$| is positive definite. Note that the identities given in (2.1) are formal in nature; however, for elements of the Banach algebra |$ \mathbb{R} ^ {n \times n} $|⁠, the identities given in (2.1) hold for all |$X\in \mathbb{R}^{n\times n}$|⁠, |$Y\in \mathbb{R}^{n\times n}$|⁠. Consider the following Cauchy problem for a non-linear impulsive differential equation (IDE): $$\begin{equation} \begin{gathered} \dot w(t)=\gamma_1(t)w(t)+f(t)w^{1+\alpha}(t),\quad t\ne k\theta,\quad w(0+0)=w_0>0,\\ w(t+0)=\beta_0w(t),\quad t=k\theta, \end{gathered} \end{equation}$$(2.2) where |$w\in \mathbb{R}$|⁠, |$\gamma _1\,:\,\mathbb{R}\to \mathbb{R}_+$|⁠, |$f\,:\,\mathbb{R}\to \mathbb{R}_+$| are piecewise continuous functions, |$\beta _0\in (0,1)$|⁠, |$\theta>0$|⁠, |$\alpha>0$| are real constants. Definition 1 The zero solution |$ w = 0 $| of IDE (2.2) is said to be (1) Lyapunov stable, if for any |$ \varepsilon> 0 $| there exists |$ \delta = \delta (\varepsilon )> 0 $| such that condition |$0<w_0<\delta $| implies that the solution |$ w (t) $| exists globally on |$[0,+\infty )$| and |$0<w(t)<\varepsilon $| for all |$t\ge 0$|⁠; (2) asymptotically stable, if it is stable and |$w(t)\to 0$| for |$t\to +\infty $|⁠. Lemma 2.1 Let |$\varrho _k=\int \limits _{k\theta }^{(k+1)\theta }f(\tau )\,\mathrm{d}\tau $|⁠, |$k\in \mathbb Z_+$|⁠. Suppose that there exist constants |$\vartheta \in (0,1)$| such that $$\begin{equation} \begin{gathered} \sup\limits_{k\in\mathbb Z_+}\int\limits_{k\theta}^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s<+\infty, \end{gathered} \end{equation}$$(2.3) $$\begin{equation} \begin{gathered} \ln\beta_0+\lim\sup\limits_{k\to\infty}\frac{1}{k}\int\limits_0^{k\theta}\gamma_1(s)\,\mathrm{d}s<\frac{\ln\vartheta}{\alpha}<0, \end{gathered} \end{equation}$$(2.4) $$\begin{align} \begin{gathered} \mu=\inf\limits_{k\in\mathbb Z_+}\beta_0^{-k\alpha}\vartheta^{k}\varrho_k^{-1}e^{-\alpha\int\limits_0^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s}>0. \end{gathered} \end{align}$$(2.5) Then any solution of IDE (2.2) with the initial condition $$\begin{equation*} w(0+0)=w_0,\quad 0<w_0<(\alpha^{-1}(1-\vartheta)\mu)^{1/\alpha} \end{equation*}$$ is extendable to the interval |$[0,+\infty )$| and $$\begin{equation*} \begin{gathered} \lim\limits_{t\to\infty}w(t)=0. \end{gathered} \end{equation*}$$ Further, the solution |$ w = 0 $| of IDE (2.2) is asymptotically stable. Proof. Let |$0<w_0<(\alpha ^{-1}(1-\vartheta )\mu )^{1/\alpha }$|⁠. Consider the solution |$ w (t) $| of IDE (2.2) with the initial condition |$w(0+0)=w_0$|⁠. Let |$[0,\omega )$| be a maximal interval of existence of the solution |$w(t)$|⁠. Suppose that |$\omega <+\infty $|⁠. Then there exists |$k^*\in \mathbb Z_+$| such that |$\omega \in [k^*\theta , (k^*+1)\theta )$|⁠. Denote |$w_k=w(k\theta +0)$|⁠, |$k=0,\dots ,k^*$|⁠, then $$\begin{equation} \begin{gathered} w(t)=e^{\int\limits_{k\theta}^t\gamma_1(s)\,\mathrm{d}s}w_k\left[1-\alpha w_k^{\alpha}\int\limits_{k\theta}^t f(\tau)e^{\alpha\int\limits_{k\theta}^{\tau}\gamma_1(s)\,\mathrm{d}s}\,\mathrm{d}\tau\right]^{-1/\alpha}. \end{gathered} \end{equation}$$(2.6) The last representation is valid for all |$t\in (k\theta ,\omega ]$|⁠, |$0\le k<k^*$|⁠, and for all |$t\in (k^*\theta ,\omega ]$|⁠. Hence, the following inequality holds: $$\begin{equation*} \begin{gathered} 1-\alpha w_{k^*}^{\alpha}\int\limits_{k^*\theta}^t f(\tau)e^{\alpha\int\limits_{k^*\theta}^{\tau}\gamma_1(s)\,\mathrm{d}s}\,\mathrm{d}\tau>0. \end{gathered} \end{equation*}$$ We introduce the following notations $$\begin{equation*} \begin{gathered} \delta_k=e^{\int\limits_{k\theta}^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s},\quad \eta_k=(\beta_0\vartheta^{-1/\alpha})^{k}e^{\int\limits_0^{k\theta}\gamma_1(s)\,\mathrm{d}s},\quad k\in\mathbb Z_+. \end{gathered} \end{equation*}$$ Using the mathematical induction, we will show that condition |$0<w_0<(\alpha ^{-1}(1-\vartheta )\mu )^{1/\alpha }$| implies the estimate |$0<w_k\le \eta _k w_0$| for all |$0\le k\le k^*$|⁠. For |$k=0$| the assertion is obvious. Suppose that the assertion also holds for a positive integer number |$0<k<k^*$|⁠, i.e., |$0<w_k\le \eta _k w_0$|⁠. Then $$\begin{equation*} \begin{gathered} 1-\alpha w_k^{\alpha}\delta_k^{\alpha}\varrho_k\ge 1-\alpha\eta_k^{\alpha}w_0^{\alpha}\delta_k^{\alpha}\varrho_k\ge 1-(1-\vartheta)\mu(\eta_k\delta_k)^{\alpha}\varrho_k\\ \ge 1-(1-\vartheta)\mu\beta_0^{\alpha k}\vartheta^{1-k}e^{\alpha\int\limits_0^{(k+1)\theta}\gamma_1(s)\,ds}\varrho_k\ge 1-(1-\vartheta)=\vartheta>0. \end{gathered} \end{equation*}$$ Hence, for all |$t\in (k\theta ,(k+1)\theta ]$|⁠, we obtain $$\begin{align*} \begin{gathered} 1-\alpha w_k^{\alpha}\int\limits_{k\theta}^t f(\tau)e^{\alpha\int\limits_{k\theta}^{\tau}\gamma_1(s)\,\mathrm{d}s}\,\mathrm{d}\tau\ge 1-\alpha w_k^{\alpha}\delta_k^{\alpha}\varrho_k\ge\vartheta>0. \end{gathered} \end{align*}$$ Integrating the comparison equation (2.2), we find $$\begin{equation} \begin{gathered} w_{k+1}=\beta_0e^{\int\limits_{k\theta}^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s}w_k\left[1-\alpha w_k^{\alpha}\int\limits_{k\theta}^{(k+1)\theta} f(\tau)e^{\alpha\int\limits_{k\theta}^{\tau}\gamma_1(s)\,\mathrm{d}s}\,\mathrm{d}\tau\right]^{-1/\alpha}\\ \le \beta_0\delta_k w_k\Big[1-\alpha w_k^{\alpha}\delta_k^{\alpha}\varrho_k\Big]^{-1/\alpha} \le\beta_0\vartheta^{-1/\alpha}\delta_kw_k\le\eta_{k+1}w_0, \end{gathered} \end{equation}$$(2.7) which proves the step of induction. Since $$\begin{equation*} \begin{gathered} 1-\alpha w_{k^*}^{\alpha}\int\limits_{k^*\theta}^t f(\tau)e^{\alpha\int\limits_{k^*\theta}^{\tau}\gamma_1(s)\,\mathrm{d}s}\,\mathrm{d}\tau\ge 1-\alpha w_{k^*}^{\alpha}\delta_{k^*}^{\alpha}\varrho_{k^*}\ge\vartheta>0,\quad t\in(k^*\theta,\omega], \end{gathered} \end{equation*}$$ then the solution |$ w (t) $| exists for all |$t\in (k^*\theta ,(k^*+1)\theta ]$|⁠. Therefore, |$\omega =+\infty $| and |$k^*=\infty $|⁠. In other words, the estimate |$0<w_k\le \eta _k w_0$| is valid for all |$k\in \mathbb Z_+$|⁠. From (2.6) it follows that, for all |$ t \in (k \theta , (k + 1) \theta ] $|⁠, $$\begin{equation*} \begin{gathered} w(t)\le e^{\int\limits_{k\theta}^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s}\eta_k\vartheta^{-1/\alpha}w_0. \end{gathered} \end{equation*}$$ Condition (2.4) yields that |$\lim \limits _{k\to \infty }\eta _k=0$|⁠. Taking into account the condition (2.3), we obtain |$\lim \limits _{t\to +\infty }w(t)=0$|⁠. The stability of the solution |$ w = 0 $| of the comparison equation (2.2) is obvious. Hence, the proof of Lemma 2.1 is completed. 3. Problem statement Consider a family of non-linear controlled systems: $$\begin{equation} \dot x(t)=\left(A(t)+\varDelta A(t,x,u)\right) x(t)+\left(B(t)+\varDelta B(t,x,u)\right) u, \end{equation}$$(3.1) where |$x\in \mathbb{R}^n$| is a state vector, |$u\in \mathbb{R}^m$| is a control vector. We assume that the mappings |$A:\mathbb{R}\rightarrow \mathbb{R}^{n\times n}$| and |$B:\mathbb{R}\rightarrow \mathbb{R}^{n\times m}$| are piecewise continuous and |$ \theta $|-periodic, that is, |$A(t+\theta )=A(t)$|⁠, |$B(t+\theta )=B(t)$| for all |$t\in \mathbb{R}$|⁠. It is assumed that the mappings |$\, \varDelta A:\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^m\rightarrow \mathbb{R}^{n\times n}$| and |$\,\varDelta B:\mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^m\rightarrow \mathbb{R}^{n\times m}$| are continuous and satisfy the following condition: for any piecewise continuous |$\theta $|-periodic function |$L\,\,:\mathbb{R}\to \mathbb{R}^{m\times n}$| the maps $$\begin{equation*} x\mapsto \varDelta A(t,x,L(t)x)x,\quad x\mapsto\varDelta B(t,x,L(t)x)L(t)x \end{equation*}$$ are locally Lipschitz in variable |$x$|⁠. We also suppose that the mappings |$ \varDelta A $| and |$ \varDelta B $| statisfy the following additional assumptions: there exist locally integrable functions |$\alpha _i:\mathbb{R}\rightarrow \mathbb{R}_+$|⁠, |$\beta _i:\mathbb{R}\rightarrow \mathbb{R}_+$|⁠, |$i=1, 2$|⁠, such that for all |$(t,x,u)\in \mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^m$|⁠, the following estimates hold: $$\begin{align} \begin{gathered} \|\varDelta A(t,x,u)\|\leq \alpha_1(t)(\|x\|+\|u\|)^{\alpha} + \beta_1(t),\\ \|\varDelta B(t,x,u)\|\leq \alpha_2(t)(\|x\|+\|u\|)^{\alpha} + \beta_2(t), \end{gathered} \end{align}$$(3.2) where |$\alpha $| is a positive constant. Following Korobov & Lutsenko (2014), it should be noted that by the robust stabilization of the family of systems (3.1), we mean the construction of a common control |$ u (t, x) = K (t) x $| ensuring the asymptotic stability of the solution |$ x = 0 $| of system (3.1) and satisfying the following conditions: (1) the mapping |$K\,:\,\mathbb{R}\to \mathbb{R}^{m\times n}$| is |$ \theta $|-periodic i.e., |$K(t+\theta )=K(t)$| for all |$t\in \mathbb{R}$|⁠; (2) the function |$ K (t) $| is piecewise constant with a finite number of values. The synthesis of the control |$ u (t, x) = K (t) x $| assumes that the zero solution of system (3.1) must be asymptotically stable, and there exists the domain of attraction which is common to all systems of family (3.1) and independent of the specific choice of functional parameters |$\varDelta A(t,x,u)$| and |$\varDelta B(t,x,u)$|⁠. Thus, in addition to synthesizing the stabilizing control |$ u (t, x) $|⁠, we will consider the problem of estimating the common domain of attraction of the solution |$ x = 0 $|⁠, for the whole family of non-linear controlled systems (3.1). 4. Construction of Lyapunov function Consider a linear system of differential equations with periodic coefficients: $$\begin{equation} \begin{gathered} \dot x(t)=C(t)x(t), \end{gathered} \end{equation}$$(4.1) where |$x\in \mathbb{R}^n$|⁠, |$C\,:\,\,\mathbb{R}\to \mathbb{R}^{n\times n}$| is a piecewise continuous and |$ \theta $|-periodic mapping, i.e., |$C(t+\theta )=C(t)$|⁠. Let $$\begin{equation*} \begin{gathered} \widehat C(t,s)=\int\limits_{s}^tC(\tau)\,\mathrm{d}\tau+\frac{1}{2}\int\limits_s^t\left[C(\tau),\int\limits_s^{\tau} C(\sigma)\,\mathrm{d}\sigma\right]\,\mathrm{d}\tau \end{gathered} \end{equation*}$$ be a partial sum of the Magnus series. Let |$N$| be a fixed positive integer number, |$h=\frac{\theta }{N}$|⁠. For each |$ m = 0, \dots , N-1 $|⁠, we define the matrices $$\begin{equation*} \begin{gathered} \widetilde C_m=\frac{1}{h}\widehat C((m+1)h,mh),\quad \widehat{C}_{m}(t)=\widehat{C}(t,mh) \end{gathered} \end{equation*}$$ and positive numbers |$a_m$|⁠, |$b_m$| and |$c_m$| such that, for |$ t \in (mh, (m + 1) h] $|⁠, the following inequalities hold: $$\begin{align} \begin{gathered} \|C(t)\|\le a_m,\quad \left\|[C(t),\int\limits_{mh}^{t} C(\sigma)\,\mathrm{d}\sigma]\right\|\le b_m(t-mh),\\ \|[C(t),\widehat{C}_m^2(t)]\|\le c_m(t-mh)^2. \end{gathered} \end{align}$$(4.2) Let |$P_0$| be a certain symmetric positive definite matrix. For each |$ m = 0,1, \dots , N-1 $|⁠, we define inductively a symmetric positive definite matrix |$ P_m $| with $$\begin{equation*} \begin{gathered} P_{m+1}=e^{-h\widetilde{ C}_m^{\operatorname{T\,}}}P_me^{-h\widetilde{ C}_m}. \end{gathered} \end{equation*}$$ We choose a Lyapunov function in the form |$v(t,x)=x^{\operatorname{T\,}} P(t)x $|⁠, where the function |$ P (t) $| is piecewise differentiable and |$ \theta $|-periodic symmetric positive definite matrix function for all |$ t \in \mathbb{R} $|⁠, |$P\in C^1(mh,(m+1)h)$|⁠. We define this function on interval |$(0,\theta ]$| as follows: $$\begin{equation*} \begin{gathered} P(t)=e^{-\widehat{C}^{\operatorname{T\,}}_m(t)}P_me^{-\widehat{C}_m(t)},\quad t\in(mh,(m+1)h]. \end{gathered} \end{equation*}$$ The time derivative of the Lyapunov function |$ v (t, x) = x ^ {\operatorname{T\,}} P (t) x $| along the trajectories of linear system (4.1) for |$ t \in (mh, (m + 1) h) $| gives the following equality $$\begin{equation*} \begin{gathered} \frac{\mathrm{d}}{\mathrm{d}t}v(t,x)\Big|_{{(4.1)}}=x^{\operatorname{T\,}}\Big(\frac{\mathrm{d}}{\mathrm{d}t}P(t)+C^{\operatorname{T\,}}(t)P(t)+P(t)C(t)\Big)x. \end{gathered} \end{equation*}$$ To calculate |$ \frac{\mathrm{d}} {\mathrm{d}t} P (t) $|⁠, we first consider the derivative |$\frac{\mathrm{d}}{\mathrm{d}t}e^{-\widehat C_m(t)}$|⁠. Applying the chain rule, we obtain $$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t}e^{-\widehat C_m(t)}&=-\Big(\frac{\mathrm{d}}{\mathrm{d}t}\widehat C_m(t)\frac{\partial}{\partial X}\Big)e^{X}\Big|_{X=-\widehat C_m(t)}\\&= -\left(\left(C(t)+\frac{1}{2}\left[C(t),\int\limits_{mh}^t C(s)\,\mathrm{d}s\right]\right)\frac{\partial}{\partial X}\right)e^{X}\Big|_{X=-\widehat C_m(t)}. \end{align*}$$ Taking into account the Hausdorff identity, we get $$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t}e^{-\widehat C_m(t)}&=-e^{-\widehat C_m(t)}\left\{C(t)+\frac{1}{2}\left[C(t),\int\limits_{mh}^t C(s)\,\mathrm{d}s\right],\frac{e^X-\operatorname{id\,}}{X}\right\}\Big|_{X=-\widehat C_m(t)}\\&= -e^{-\widehat C_m(t)}\left(C(t)+\frac{1}{2}\left[C(t),\int\limits_{mh}^t C(s)\,\mathrm{d}s\right]\right.\\&\quad\left.+ \sum\limits_{k=1}^{\infty}\frac{(-1)^k}{(k+1)!}\left\{C(t) +\frac{1}{2}\left[C(t),\int\limits_{mh}^t C(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right)\\ &=-e^{-\widehat C_m(t)}\left(C(t)-\frac{1}{4}\left[C(t),\int\limits_{mh}^t\left[C(\tau),\int\limits_{mh}^{\tau}C(\sigma)\,\mathrm{d}\sigma\right]\,\mathrm{d}\tau\right]\right.\\&\quad\left.+ \sum\limits_{k=2}^{\infty}\frac{(-1)^k}{(k+1)!}\left\{C(t),\widehat{C}_m^k(t)\right\}+\frac{1}{2}\sum\limits_{k=1}^{\infty} \frac{(-1)^k}{(k+1)!}\left\{\left[C(t),\int\limits_{mh}^tC(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right). \end{align*}$$ Thus, for |$t\in (mh,(m+1)h]$|⁠, we obtain $$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t}P(t)&=-\left(C(t)-\frac{1}{4}\left[C(t),\int\limits_{mh}^t\left[C(\tau),\int\limits_{mh}^{\tau}C(\sigma)\,\mathrm{d}\sigma\right]\,\mathrm{d}\tau\right]\right.\\&\quad\left.+ \sum\limits_{k=2}^{\infty}\frac{(-1)^k}{(k+1)!}\left\{C(t),\widehat{C}_m^k(t)\right\}+\frac{1}{2}\sum\limits_{k=1}^{\infty} \frac{(-1)^k}{(k+1)!}\left\{\left[C(t),\int\limits_{mh}^tC(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right)^{\operatorname{T\,}}P(t)\\ &\quad- P(t)\left(C(t)-\frac{1}{4}\left[C(t),\int\limits_{mh}^t\left[C(\tau),\int\limits_{mh}^{\tau}C(\sigma)\,\mathrm{d}\sigma\right]\,\mathrm{d}\tau\right]\right.\\&\quad\left.+ \sum\limits_{k=2}^{\infty}\frac{(-1)^k}{(k+1)!}\left\{C(t),\widehat{C}_m^k(t)\right\}+\frac{1}{2}\sum\limits_{k=1}^{\infty} \frac{(-1)^k}{(k+1)!}\left\{\left[C(t),\int\limits_{mh}^tC(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right). \end{align*}$$ Note that $$\begin{equation*} \begin{gathered} \|\widehat C_m(t)\|\le a_m(t-mh)+\frac{1}{4}b_m(t-mh)^2. \end{gathered} \end{equation*}$$ Taking into account inequalities in (4.2), it is not difficult to obtain the following estimates for |$t\in (mh,(m+1)h]$|⁠: $$\begin{equation*} \begin{gathered} \left\|\left[C(t),\int\limits_{mh}^t[C(\tau),\int\limits_{mh}^{\tau}C(\sigma)\,\mathrm{d}\sigma]\,\mathrm{d}\tau\right]\right\|\le a_mb_m(t-mh)^2, \end{gathered} \end{equation*}$$ $$\begin{equation*} \begin{gathered} \big\|[C(t),\widehat{C}_m^{k}(t)]\big\|\le c_m(t-mh)^k\left(2a_m+\frac{1}{2}b_m(t-mh)\right)^{k-2}, \end{gathered} \end{equation*}$$ $$\begin{align*} \begin{gathered} \left\|\left\{\left[C(t),\int\limits_{mh}^tC(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right\|\le b_m(t-mh)^{k+1}\left(2a_m+\frac{1}{2}b_m(t-mh)\right)^{k}. \end{gathered} \end{align*}$$ Hence, we obtain $$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t}v(t,x)\Big|_{{(4.1)}}&=-x^{\operatorname{T\,}}\left(\left(-\frac{1}{4}\left[C(t),\int\limits_{mh}^t\left[C(\tau),\int\limits_{mh}^{\tau}C(\sigma)\,\mathrm{d}\sigma\right]\,\mathrm{d}\tau\right]\right.\right.\\&\quad\left.+ \sum\limits_{k=2}^{\infty}\frac{(-1)^k}{(k+1)!}\left\{C(t),\widehat{C}_m^k(t)\right\}+\frac{1}{2}\sum\limits_{k=1}^{\infty} \frac{(-1)^k}{(k+1)!}\left\{\left[C(t),\int\limits_{mh}^tC(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right)^{\operatorname{T\,}}P(t)\\ \end{align*}$$ $$\begin{align*} &\quad+P(t)\left(-\frac{1}{4}\left[C(t),\int\limits_{mh}^t\left[C(\tau),\int\limits_{mh}^{\tau}C(\sigma)\,\mathrm{d}\sigma\right]\,\mathrm{d}\tau\right]\right.\\&\quad\left.\left.+ \sum\limits_{k=2}^{\infty}\frac{(-1)^k}{(k+1)!}\left\{C(t),\widehat{C}_m^k(t)\right\}+\frac{1}{2}\sum\limits_{k=1}^{\infty} \frac{(-1)^k}{(k+1)!}\left\{\left[C(t),\int\limits_{mh}^tC(s)\,\mathrm{d}s\right],\widehat{C}_m^k(t)\right\}\right)\right)x\\ &\le 2\|P^{-1/2}(t)\|\|P^{1/2}(t)\|\left(\frac{1}{4}a_mb_m(t-mh)^2+\sum\limits_{k=2}^{\infty}\frac{c_m(t-mh)^k(2a_m+\frac{1}{2}b_m(t-mh))^{k-2}}{(k+1)!}\right.\\&\quad\left.+ \frac{1}{2}\sum\limits_{k=1}^{\infty}\frac{b_m(t-mh)^{k+1}(2a_m+\frac{1}{2}b_m(t-mh))^{k}}{(k+1)!}\right)v(t,x). \end{align*}$$ Also, the following estimates hold: $$\begin{equation} \begin{gathered} \|P^{1/2}(t)\|=\lambda_{\max}^{1/2}(P(t))=\|P(t)\|^{1/2}\le e^{a_m(t-mh)+\frac{1}{4}b_m(t-mh)^2}\|P_m\|^{1/2}, \end{gathered} \end{equation}$$(4.3) $$\begin{equation} \begin{gathered} \|P^{-1/2}(t)\|=\lambda_{\max}^{1/2}(P^{-1}(t))=\lambda_{\min}^{-1/2}(P(t))\le e^{a_m(t-mh)+\frac{1}{4}b_m(t-mh)^2}\lambda_{\min}^{-1/2}(P_m). \end{gathered} \end{equation}$$(4.4) Hence, we get $$\begin{align} \frac{\mathrm{d}}{\mathrm{d}t}v(t,x)\Big|_{{(4.1)}}&\le 2\frac{\|P_m\|^{1/2}}{\lambda_{\min}^{1/2}(P_m)}e^{2a_m(t-mh)+\frac{1}{2}b_m(t-mh)^2}\left(\frac{1}{4}a_mb_m(t-mh)^2\vphantom{\frac{1}{2}\sum\limits_{k=1}^{\infty}\frac{b_m(t-mh)^{k+1}(2a_m+\frac{1}{2}b_m(t-mh))^{k}}{(k+1)!}}\right.\nonumber\\ &\quad+\sum\limits_{k=2}^{\infty}\frac{c_m(t-mh)^k(2a_m+\frac{1}{2}b_m(t-mh))^{k-2}}{(k+1)!}\nonumber\\&\quad\left.+ \frac{1}{2}\sum\limits_{k=1}^{\infty}\frac{b_m(t-mh)^{k+1}(2a_m+\frac{1}{2}b_m(t-mh))^{k}}{(k+1)!}\right)v(t,x). \end{align}$$(4.5) We now introduce the function |$ \gamma (t) $|⁠, which is defined on each half-open interval |$ (mh, (m + 1) h] $|⁠, |$m=0,\dots ,N-1$|⁠, by the formula: $$\begin{align*} \gamma(t)&=2\frac{\|P_m\|^{1/2}}{\lambda_{\min}^{1/2}(P_m)}e^{2a_m(t-mh)+\frac{1}{2}b_m(t-mh)^2}\left(\frac{1}{4}a_mb_m(t-mh)^2\vphantom{+ \frac{1}{2}\sum\limits_{k=1}^{\infty}\frac{b_m(t-mh)^{k+1}(2a_m+\frac{1}{2}b_m(t-mh))^{k}}{(k+1)!}}\right.\\ &\quad+\sum\limits_{k=2}^{\infty}\frac{c_m(t-mh)^k(2a_m+\frac{1}{2}b_m(t-mh))^{k-2}}{(k+1)!}\\&\quad\left.+ \frac{1}{2}\sum\limits_{k=1}^{\infty}\frac{b_m(t-mh)^{k+1}(2a_m+\frac{1}{2}b_m(t-mh))^{k}}{(k+1)!}\right) \end{align*}$$ and extend this function to the whole axis |$ \mathbb{R} $|⁠, assuming that it is |$ \theta $|-periodic. Then, we can conclude that $$\begin{equation} \begin{gathered} \frac{\mathrm{d}}{\mathrm{d}t}v(t,x)\Big|_{{(4.1)}}\le \gamma(t) v(t,x). \end{gathered} \end{equation}$$(4.6) 5. Main result In this section, we apply the results of Section 4 to the synthesis of control, which stabilizes a family of non-linear controlled system (3.1). Let $$\begin{equation*} \begin{gathered} \widehat{A}(t,s)=\int\limits_s^t A(\tau)\mathrm{d}\tau+\frac12\int\limits_s^t\left[A(\tau), \int\limits_s^\tau A(\sigma)\mathrm{d}\sigma\right]\mathrm{d}\tau \end{gathered} \end{equation*}$$ and |$N$| be a fixed positive integer, |$h=\frac{\theta }{N}$|⁠. For each |$ m = 0, \dots , N-1 $|⁠, we define the constant matrices: $$\begin{equation*} \begin{gathered} A_m=\frac1h\widehat{A}((m+1)h,mh),\quad B_m=\frac1h\int\limits_{mh}^{(m+1)h}B(s)\mathrm{d}s. \end{gathered} \end{equation*}$$ Assumption 5.1 Assume that, for any |$m=0,1,\ldots ,N-1$|⁠, the pairs of matrices |$ \left (A_m, B_m \right ) $| are stabilizable, i.e., there exists a linear feedback |$ u (y) = K_my $| such that the following linear differential system with constant coefficients $$\begin{equation*} \begin{gathered} \dot y(t)=\left(A_m + B_mK_m\right)y(t) \end{gathered} \end{equation*}$$ is asymptotically stable. Remark If the pair of matrices |$ (A_m, B_m) $| is controllable, then the matrix |$ K_m $| can be chosen explicitly (see Korobov & Lutsenko, 2014, Lemma 1) such that $$\begin{equation*} \begin{gathered} K_m=-B^{\operatorname{T\,}}_mN_{\lambda_m,m}^{-1}(t_{1m}),\quad N_{\lambda_m,m}(t_{1m})=\int\limits_{0}^{t_{1m}}W_m(s)\,\mathrm{d}s,\\ W_m(t)=e^{-2\lambda_m t}e^{-A_mt}B_mB^{\operatorname{T\,}}_me^{-A^{\operatorname{T\,}}_mt}, \end{gathered} \end{equation*}$$ where |$\lambda _m$| and |$t_{1m}$| are arbitrary numbers satisfying the conditions |$\lambda _m>0$|⁠, |$0<t_{1m}\le +\infty $|⁠, and for |$ t_ {1m} =+\infty $| the number |$\lambda _m$| is chosen from the condition $$\begin{equation*} \begin{gathered} \lambda_m>\max\limits_{\lambda\in\sigma(A_m)}\operatorname{Re\,}(-\lambda). \end{gathered} \end{equation*}$$ Assumption 5.1 allows us to construct the feedback function for the original non-autonomous system (3.1). Let |$k\in \mathbb Z$|⁠, then we put $$\begin{equation*} \begin{gathered} K(t)=K_m,\quad t\in(k\theta+mh,k\theta+(m+1)h],\quad m=0,\dots,N-1,\\ u(t,x)=K(t)x. \end{gathered} \end{equation*}$$ Next, we establish the conditions under which this feedback stabilizes the equilibrium |$ x = 0 $| of the family of non-linear control systems (3.1), and also estimate the domain of attraction of |$ x = 0 $|⁠. Let $$\begin{equation*} \begin{gathered} C(t)=A(t)+B(t)K(t), \end{gathered} \end{equation*}$$ $$\begin{equation*} \begin{gathered} \widehat{C}(t,s)=\int\limits_s^t C(\tau)\mathrm{d}\tau + \frac{1}{2}\int\limits_s^t\left[C(\tau),\int\limits_s^\tau C(\sigma)\mathrm{d}\sigma\right]\mathrm{d}\tau. \end{gathered} \end{equation*}$$ For each |$ m = 0, \dots , N-1 $|⁠, we define the matrices $$\begin{equation*} \begin{gathered} \widetilde C_m=\frac{1}{h}\widehat C((m+1)h,mh),\quad \widehat{C}_{m}(t)=\widehat{C}(t,mh). \end{gathered} \end{equation*}$$ Let |$\varPhi =e^{h\widetilde{ C}_{N-1}}\dots e^{h\widetilde{ C}_0} $|⁠. If |$r_{\sigma }(\varPhi )<1$|⁠, then there exists a symmetric positive definite matrix |$ P_0 $| satisfying the LMI $$\begin{equation*} \begin{gathered} \varPhi^{\operatorname{T\,}}P_0\varPhi-P_0\prec 0. \end{gathered} \end{equation*}$$ Let us define inductively symmetric positive definite matrices |$P_m$|⁠, |$m=0,\dots ,N-1$|⁠, $$\begin{equation*} \begin{gathered} P_{m+1}=e^{-h\widetilde{ C}_m^{\operatorname{T\,}}}P_me^{-h\widetilde{ C}_m}. \end{gathered} \end{equation*}$$ Assumption 5.2 Assume that there exist positive numbers |$a_m$|⁠, |$b_m$| and |$c_m$| such that, for |$t\in (mh,(m+1)h)$|⁠, the following inequalities hold: $$\begin{equation*} \begin{gathered} \|C(t)\|\le a_m,\quad \left\|[C(t),\int\limits_{mh}^{t} C(\sigma)\,\mathrm{d}\sigma]\right\|\le b_m(t-mh),\\ \|[C(t),\widehat{C}_m^2(t)]\|\le c_m(t-mh)^2. \end{gathered} \end{equation*}$$ We now introduce the following functions: $$\begin{align*} \gamma_1(t)&=\gamma(t)+(\beta_1(t)+\beta_2(t)\|K_m\|)e^{2a_m(t-mh)+\frac{1}{2}b_m(t-mh)^2}\|P_m\|^{1/2}\lambda_{\min}^{-1/2}(P_m),\\ f(t)&=\|P_m\|^{1/2}\lambda_{\min}^{-(1+\alpha)/2}(P_m)e^{(2+\alpha)(a_m(t-mh)+\frac{1}{4}b_m(t-mh)^2)}\\ &\quad\times(\alpha_1(t)+\alpha_2(t)\|K_m\|)(1+\|K_m\|)^{\alpha},\quad t\in(k\theta+mh,k\theta+(m+1)h] \end{align*}$$ and sequence of numbers |$\varrho _k=\int \limits _{k\theta }^{(k+1)\theta }f(\tau )\,\mathrm{d}\tau $|⁠, |$k\in \mathbb Z_+$|⁠. Theorem 5.1 Let Assumptions 5.1 and 5.2 be satisfied for the control |$ u (t, x) = K (t) x $|⁠, and there exist a positive integer |$ N $| such that |$r_{\sigma }(\varPhi )<1$|⁠. In addition, we assume that for a certain number |$ \vartheta \in (0,1) $| the following inequalities hold: $$\begin{equation} \begin{gathered} \sup\limits_{k\in\mathbb Z_+}\int\limits_{k\theta}^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s<+\infty, \end{gathered} \end{equation}$$(5.1) $$\begin{equation} \begin{gathered} \ln\lambda_{\max}(P_N^{-1}P_0)+\lim\sup\limits_{k\to\infty}\frac{1}{k}\int\limits_0^{k\theta}\gamma_1(s)\,\mathrm{d}s<\frac{2\ln\vartheta}{\alpha}<0, \end{gathered} \end{equation}$$(5.2) $$\begin{equation} \begin{gathered} \mu=\inf\limits_{k\in\mathbb Z_+}\left\{(\lambda_{\max}(P_N^{-1}P_0))^{-k\alpha/2}\vartheta^{k-1}\varrho_k^{-1}e^{-\frac{\alpha}{2}\int\limits_0^{(k+1)\theta}\gamma_1(s)\,\mathrm{d}s}\right\}>0. \end{gathered} \end{equation}$$(5.3) Then the control |$ u (t, x) = K (t) x $| stabilizes the family of non-linear controlled systems (3.1). Furthermore, the domain of attraction of equilibrium |$ x = 0 $| for all systems of the family contains the following ellipsoid: $$\begin{equation*} \begin{gathered} \varOmega=\{x\in\mathbb{R}^n\,\,:\,\,x^{\operatorname{T\,}}P_0x<(2\alpha^{-1}(1-\vartheta)\mu)^{2/\alpha}\}. \end{gathered} \end{equation*}$$ Proof. Consider the Lyapunov function |$ v (t, x) = x ^ {\operatorname{T\,}} P (t) x $|⁠, where $$\begin{equation*} \begin{gathered} P(t)=e^{-\widehat{C}_m^{\operatorname{T\,}}(t)}P_m e^{-\widehat{C}_m(t)},\quad t\in(k\theta+mh,k\theta+(m+1)h]. \end{gathered} \end{equation*}$$ Let us estimate the derivative of this Lyapunov function along the trajectories of non-linear controlled system (3.1) for |$t\in (k\theta +mh,k\theta +(m+1)h]$|⁠. Then, we can obtain $$\begin{align*} \frac{\mathrm{d}}{\mathrm{d}t}v(t,x(t))\Big|_{{(3.1)}}&\le \gamma(t)v(t,x(t))+ 2x^{\operatorname{T\,}}(t)P(t)\varDelta A(t,x(t),u(t,x))x(t)\\&\quad+ 2x^{\operatorname{T\,}}(t)P(t)\varDelta B(t,x(t),u(t,x))u(t,x)\\ &\le (\gamma(t)+(\beta_1(t)+\beta_2(t)\|K_m\|)\|P^{1/2}(t)\|\|P^{-1/2}(t)\|)v(t,x(t))\\ &\quad+\|P^{1/2}(t)\|\|P^{-1/2}(t)\|^{1+\alpha}(\alpha_1(t)+\alpha_2(t)\|K_m\|)(1+\|K_m\|)^{\alpha} v^{1+\frac{\alpha}{2}}(t,x(t)). \end{align*}$$ Taking into account estimates (4.3) and (4.4), we obtain the following differential inequality for the function |$ v (t, x) $| provided that |$t\in (k\theta ,(k+1)\theta ]$|⁠: $$\begin{equation*} \begin{gathered} \frac{\mathrm{d}}{\mathrm{d}t}v(t,x(t))\Big|_{{(3.1)}}\le \gamma_1(t)v(t,x(t))+f(t)v^{1+\frac{\alpha}{2}}(t,x(t)). \end{gathered} \end{equation*}$$ At time instants |$ t = k \theta $|⁠, we obtain the estimate: $$\begin{equation*} \begin{gathered} v(t+0,x(t+0))\le\lambda_{\max}(P_{N}^{-1}P_0)v(t,x(t)). \end{gathered} \end{equation*}$$ Applying the comparison principle (see Chatterjee & Liberzon, 2006), we get $$\begin{equation*} \begin{gathered} v(t,x(t))\le w(t),\quad t\in[0,\omega), \end{gathered} \end{equation*}$$ where |$w(t)$| is the solution of the Cauchy problem for the IDE: $$\begin{equation} \begin{gathered} \dot w(t)=\gamma_1(t)w(t)+f(t)w^{1+\frac{\alpha}{2}}(t),\quad t\ne k\theta,\quad w(0+0)=v(0+0,x_0)=x_0^{\operatorname{T\,}}P_0x_0,\\ w(t+0)=\lambda_{\max}(P_{N}^{-1}P_0)w(t),\quad t=k\theta, \end{gathered} \end{equation}$$(5.4) and |$[0,\omega )$| is the maximum interval of existence of the solution |$w(t)$|⁠. From the conditions (5.1), (5.2) and (5.3) of Theorem 4.1 and Lemma 1.1, it follows that if |$ x_0 \in \varOmega $|⁠, then |$\omega =+\infty $| and the solution |$ w = 0 $| of IDE (comparison equation) (5.4) is asymptotically stable, and the domain of attraction of the solution |$ w = 0 $| contains interval |$(0,(2\alpha ^{-1}(1-\vartheta )\mu )^{2/\alpha })$|⁠. If |$\varepsilon>0$|⁠, then, from the stability of the solution |$ w = 0 $| of IDE (comparison equation) (5.4), it follows that there exists a positive number |$ \delta _1 = \delta _1 (\varepsilon ) $| such that inequality |$0<w_0<\delta _1$| implies |$0<w(t)<\varepsilon ^2\lambda _{\min }(P_0)$|⁠. If |$\delta =\sqrt{\frac{\delta _1}{\lambda _{\max }(P_0)}}$| and |$\|x_0\|<\delta $|⁠, then $$\begin{equation*} \begin{gathered} 0<w_0=v(0+,x_0)=x_0^{\operatorname{T\,}}P_0x_0\le\lambda_{\max}(P_0)\|x_0\|^2<\delta_1. \end{gathered} \end{equation*}$$ Hence, for all |$t>0$|⁠, we obtain $$\begin{equation*} \begin{gathered} \lambda_{\min}(P_0)\|x(t)\|^2\le v(t,x(t))\le w(t)\le \varepsilon^2\lambda_{\min}(P_0). \end{gathered} \end{equation*}$$ So, it follows that |$ \| x (t) \| <\varepsilon $| for all |$ t> 0 $|⁠, which proves the stability of the solution |$ x = 0 $| of the family of non-linear controlled systems (3.1). If |$x_0\in \varOmega $|⁠, then |$w_0=w(0)<(2\alpha ^{-1}(1-\vartheta )\mu )^{2/\alpha }$| and |$w(t)\to 0$| for |$t\to +\infty $|⁠. From inequality $$\begin{equation*} \begin{gathered} \|x(k\theta)\|\le\sqrt{\frac{w(k\theta+0)}{\lambda_{\min}(P_0)}}, \end{gathered} \end{equation*}$$ it follows that |$\|x(k\theta )\|\to 0$| for |$k\to \infty $|⁠. Applying the theorem on continuous dependence of solutions of differential equations on initial conditions, we conclude that |$\|x(t)\|\to 0$| for |$t\to +\infty $|⁠. Hence, Theorem 5.1 is proved. Remark If |$\beta _1(t)=\beta _2(t)=0$| and matrix |$C(t)$| satisfies the Lappo-Danilevsky condition, i.e., |$\Big [C(t),\int \limits _{mh}^t C(s)\,\mathrm{d}s\Big ]=0$| for all |$t\in [mh,(m+1)h]$|⁠, then |$\gamma _1(t)=0$|⁠, and conditions (5.1) and (5.2) reduce to |$\lambda _{\max }(P_N^{-1}P_0)<\vartheta ^{2/\alpha }$|⁠. 6. Numerical example Consider a family of non-linear controlled systems (3.1) with the following parameters: $$\begin{equation*} A(t)=A_0+\rho(G_1\cos\omega t+G_2\sin\omega t)\in\mathbb{R}^{2\times 2}, \quad B(t)\equiv B_0\in\mathbb{R}^{2\times 2},\quad\alpha=1, \end{equation*}$$ where |$\rho \in \mathbb{R}$|⁠, |$A_0$|⁠, |$B_0$|⁠, |$\det B_0\ne 0$|⁠, |$G_1$| and |$G_2$| are constant matrices given by $$\begin{equation*} \begin{gathered} G_1= \left( \begin{array}{@{}cc@{}} 1&0\\ 0&-1 \end{array}\right),\quad G_2= \left( \begin{array}{@{}cc@{}} 0&-1\\ -1&0 \end{array}\right). \end{gathered} \end{equation*}$$ The matrices |$G_0$|⁠, |$G_1$|⁠, |$G_2$| satisfy the commutation relations as the following: $$\begin{equation*} [G_0,G_2]=-2G_1,\quad [G_0,G_1]=2G_2,\quad [G_1,G_2]=-2G_0, \end{equation*}$$ where $ G_0= \left ( \begin{array}{@{}cc@{}} 0&1\\ -1&0 \end{array}\right ) $ ⁠. For the functions |$\varDelta A(t,x,u)$| and |$\varDelta B(t,x,u)$|⁠, we assume that $$\begin{equation} \|\varDelta A(t,x,u)\|\le\varepsilon(\|x\|+\|w\|),\quad \varDelta B(t,x,u)\equiv 0, \end{equation}$$(6.1) for all |$(t,x,u)\in \mathbb{R}\times \mathbb{R}^n\times \mathbb{R}^m$|⁠. Let |$N=1$|⁠, |$h=\theta =\frac{2\pi }{\omega }$|⁠, |$m=0$|⁠. Then the control has the form of linear feedback |$u(x)=Kx$|⁠, where |$K\in \mathbb{R}^{2\times 2}$| is a constant matrix that we define from the condition $$\begin{equation} K=-B^{-1}_0(pI+A_0), \end{equation}$$(6.2) where |$p$| is a positive constant. Consequently, it follows that $$\begin{equation*} C(t)=-pI+\rho(G_1\cos\omega t+G_2\sin\omega t). \end{equation*}$$ Let |$\chi _0(t)=t$|⁠, |$\chi _1(t)=\frac{\sin \omega t}{\omega }$|⁠, |$\chi _2(t)=\frac{1-\cos \omega t}{\omega }$|⁠. Let us calculate the following matrices: $$\begin{equation*} \begin{gathered} \widehat{C}_0(t)=-p\chi_0(t)I+\rho\chi_1(t)G_1+\rho\chi_2(t)G_2+\frac{\omega t-\sin\omega t}{\omega^2}\rho^2G_0,\\ \left[C(t),\int\limits_0^t C(\tau)\,\mathrm{d}\tau\right]=2\frac{1-\cos\omega t}{\omega}\rho^2G_0,\\ \operatorname{ad\,}_{\widehat{C}_0(t)}=\rho\chi_1(t)\operatorname{ad\,}_{G_1}+\rho\chi_2(t)\operatorname{ad\,}_{G_2}+\frac{\omega t-\sin\omega t}{\omega^2}\rho^2\operatorname{ad\,}_{G_0}, \end{gathered} \end{equation*}$$ $$\begin{align*} \widetilde{C}_0=-pI+\frac{\theta}{2\pi}\rho^2G_0. \end{align*}$$ Applying the Cauchy inequality, we obtain estimates: $$\begin{equation*} \begin{gathered} \|C(t)\|\le \sqrt{3(p^2+\rho^2)}=a_0,\\ \left\|\left[C(t),\int\limits_0^t C(\tau)\,\mathrm{d}\tau\right]\right\|\le \frac{2\rho^2}{\pi}\theta=b_0,\\ \|\operatorname{ad\,}_{\widehat{C}_0(t)}\|\le 2\sqrt{3}\rho\sqrt{\chi_1^2(t)+\chi_2^2(t)+\left(\frac{2\pi\rho}{\omega^2}\right)^2}. \end{gathered} \end{equation*}$$ Consequently, we can derive $$\begin{equation*} \begin{gathered} \|\{C(t),\widehat{C}_0^2(t)\}\|\le \|C(t)\|\|\operatorname{ad\,}_{\widehat{C}_0(t)}\|^2\le\frac{12a_0\rho^2\theta^2}{\pi^2}\left(1+\frac{\theta^2\rho^2}{4}\right)=c_0. \end{gathered} \end{equation*}$$ Then, it is not difficult to calculate $$\begin{equation*} \begin{gathered} \gamma(t)\le 2e^{2a_0\theta+\frac{1}{2}b_0\theta^2}\left(\frac{1}{4}a_0b_0\theta^2+\sum\limits_{k=2}^{\infty}\frac{c_0\theta^k(2a_0+0.5b_0\theta)^{k-2}}{(k+1)!}+ \sum\limits_{k=1}^{\infty}\frac{b_0\theta^{k+1}(2a_0+0.5b_0\theta)^{k}}{(k+1)!}\right)=\gamma^*. \end{gathered} \end{equation*}$$ Since |$\beta _1(t)=\beta _2(t)=0$|⁠, |$\alpha _1(t)=\varepsilon $|⁠, |$\alpha _2(t)=0$|⁠, then $$\begin{equation*} \begin{gathered} \gamma(t)\le\gamma^*,\quad f(t)\le e^{3(a_0\theta+\frac{1}{4}b_0\theta^2)}\varepsilon(1+\|K\|):=f^*. \end{gathered} \end{equation*}$$ Suppose that |$r_{\sigma }(\varPhi )<1$|⁠, where |$\varPhi =e^{\theta \widetilde C_0}$|⁠. Then there exists a symmetric positive definite matrix |$P_0\in \mathbb{R}^{2\times 2}$| such that $$\begin{equation*} \varPhi^{\operatorname{T\,}}P_0\varPhi-P_0\prec 0. \end{equation*}$$ Let $$\begin{equation} \begin{gathered} P_1=e^{-\theta\widetilde C_0^{\operatorname{T\,}}}P_0e^{-\theta\widetilde C_0}. \end{gathered} \end{equation}$$(6.3) Then the condition for the asymptotic stability of the equilibrium |$x=0$| is $$\begin{equation} \begin{gathered} \ln\lambda_{\max}(P_1^{-1}P_0)+\gamma^*\theta<0. \end{gathered} \end{equation}$$(6.4) If this condition is satisfied, then $$\begin{equation} \vartheta=\lambda_{\max}^{1/2}(P_1^{-1}P_0)e^{\frac{\gamma^*\theta}{2}}<1,\quad \mu=\frac{e^{-\frac{\gamma^*\theta}{2}}}{\theta f^*}. \end{equation}$$(6.5) Therefore, Theorem 5.1 implies that the common region of attraction of the equilibrium |$x=0$| for all non-linear controlled systems (3.1) contains the circle $$\begin{equation*} \begin{gathered} \varOmega=\{x\in\mathbb{R}^2\,\,:\,\,\|x\|<2(1-\vartheta)\mu\}. \end{gathered} \end{equation*}$$ In the particular case of non-linear controlled system (3.1), we choose |$A_0$| and |$B_0$| as the following: $$\begin{equation*} \begin{gathered} A_0=\left( \begin{array}{@{}cc@{}} 1&2\\ 1&1 \end{array}\right), \quad B_0=\left( \begin{array}{@{}cc@{}} 5&1\\ 1&10 \end{array}\right), \end{gathered} \end{equation*}$$ and |$\theta =0.1$|⁠, |$\rho =1$|⁠, |$\varepsilon =0.1$|⁠. Let us take |$p=0.8$|⁠. Then $$\begin{equation*} K=\left( \begin{array}{@{}cc@{}} -0.3469&-0.3714\\ -0.0653&-0.1429 \end{array}\right),\quad \widetilde C_0=\left( \begin{array}{@{}cc@{}} -0.8&0.01591\\ -0.01591&-0.8 \end{array}\right) \end{equation*}$$ and |$a_0=2.218107$|⁠, |$b_0=0.063661$|⁠, |$c_0=0.027036$|⁠, |$\gamma ^*=0.00198$|⁠, |$f^*=0.297744$|⁠. We can choose |$P_0=I$|⁠. Then |$P_1=1.17351 I$|⁠, |$\vartheta =0.923208$|⁠, |$\mu =33.582527$|⁠. Hence, we can conclude that the region of attraction of the equilibrium state |$x=0$|⁠, common to all non-linear controlled systems (3.1), contains a circle |$\varOmega =\{x\in \mathbb{R}^2\,\,:\,\,\|x\|<5.1576\}$|⁠. 7. Discussion The obtained approach for the stabilization of the family of non-linear non-autonomous controlled systems allows for an efficient numerical implementation. 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TI - Robust stabilization of non-linear non-autonomous control systems with periodic linear approximation
JF - IMA Journal of Mathematical Control and Information
DO - 10.1093/imamci/dnaa003
DA - 2021-03-16
UR - https://www.deepdyve.com/lp/oxford-university-press/robust-stabilization-of-non-linear-non-autonomous-control-systems-with-7HDroaISLk
SP - 125
EP - 142
VL - 38
IS - 1
DP - DeepDyve
ER -