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

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 M. It is shown that the solution of the LQ-optimal control problem is obtained in terms of the positive semi-definite solution S(ℓ), ℓ ∈ M, of M-coupled algebraic Riccati equations. By M-coupled we mean that the algebraic Riccati equations are coupled via an integral over a transition probability kernel G(·|·) having a density g(·|·) with respect to a σ-finite measure μ on 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.