LQ Control of Discrete-Time Jump Systems With Markov Chain in a General Borel Space

Abstract

In this technical note, it is studied the LQ-optimal control problem for discrete-time Markov jump linear systems considering the case in which the Markov chain takes values in a general Borel space ${\cal M}$ . It is shown that the solution of the LQ-optimal control problem is obtained in terms of the positive semi-definite solution $S(\ell)$ , $\ell\in {\cal M}$ , of ${\cal M}$ -coupled algebraic Riccati equations. By ${\cal M}$ -coupled we mean that the algebraic Riccati equations are coupled via an integral over a transition probability kernel ${\cal G}(\cdot\vert\cdot)$ having a density $g(\cdot\vert\cdot)$ with respect to a $\sigma$ -finite measure $\mu$ on ${\cal M}$ . It is obtained sufficient conditions, based on the concept of stochastic stabilizability and stochastic detectability, for the existence and uniqueness of this positive semi-definite solution. These results generalize previous ones in the literature, which considered only the case of the Markov chain taking values in a finite or infinite countable space.