Optimal control for continuous-time linear quadratic problems with infinite Markov jump parameters (Q2753204)

The optimal control system under the usual infinite horizon quadratic cost considered here is NEWLINE\[NEWLINE\dot x(t)=A_{\theta(t)} x(t)+ B_{\theta(t)}u(t),\, s<t<T;\;s,T\in[0,\infty);\;x(s)=x_s,\;\theta(s)= \theta_s,NEWLINE\]NEWLINE where the state vector \(x(t)\in\mathbb{C}^n\), the control input \(u(t)\in\mathbb{C}^m\), and \(A\) and \(B\) are functions of a homogeneous right-continuous Markov process \(\theta(t)\) with an infinite countable state space. A similar discrete-time \(LQ\)-optimal problems had been studied by \textit{O. L. V. Costa} and \textit{M. D. Fragoso} [IEEE Trans. Autom. Control 40, 2076--2088 (1995; Zbl 0843.93091)]. By making use of semigroup theory and a decomplexification technique under the continuous time feature, the optimal problem is then equivalent to the question of existence and uniqueness of the solution to a certain countably infinite set of coupled algebraic Riccati equations (ICARE). After defining two essential concepts of stochastic stability and stochastic detectability, the authors obtain conditions for existence and uniqueness of a positive semidefinite solution of the ICARE.NEWLINENEWLINE It is worth mentioning that when applying these results to the case of finite state space of the corresponding Markov chain, the conditions for the optimal control policy (e.g., stochastic detectability instead of observability) are somewhat relaxed [cf. Theorem 5 of \textit{Y. Ji} and \textit{H. J. Chizeck}, IEEE Trans. Autom. Control 35, 777--788 (1990; Zbl 0714.93060)].