Numerical solution for the linear-quadratic control problem of Markov jump linear systems and a weak detectability concept

A method for solving the linear quadratic problem of Markov jump linear systems is developed in this paper, relying on the assumption of weak detectability. This concept of detectability generalizes previous concepts relevant to this class of systems, and most importantly, it allows us to revisit the quadratic control problem. In the main result of the paper, we show for weakly detectable systems that the solution of the new method converges to the solution of the coupled algebraic Riccati equation if and only if the system is mean-square stabilizable.

1 Introduction

Consider the coupled algebraic Riccati equation (CARE) in the unknown :

$$\begin{equation*} A_{i}^{\prime}X_{i}+X_{i}A_{i}-X_{i}B_{i}R_{i}^{-1}B_{i}^{\prime} X_{i}+C_{\mathrm{i}}^{\prime}C_{\mathrm{i}}+\displaystyle \sum_{j=1}^{N}\lambda_{\mathrm{i}j}X_{j}=0,\ \mathrm{i}=1, \ldots, N \tag{1} \end{equation*}$$ where is given, and and are matrices that belong to given collections of real matrices and , Equation (1) arises in the linear quadratic (LQ) control problem of Markov jump linear systems (MJLS).