Linear Quadratic Optimal Control of Discrete-Time Stochastic Systems Driven by Homogeneous Markov Processes

Random terms in many natural and social science systems have distinct Markovian characteristics, such as Markov jump-taking values in a finite or countable set, and Wiener process-taking values in a continuous set. In general, these systems can be seen as Markov-process-driven systems, which can be used to describe more complex phenomena. In this paper, a discrete-time stochastic linear system driven by a homogeneous Markov process is studied, and the corresponding linear quadratic (LQ) optimal control problem for this system is solved. Firstly, the relations between the well-posedness of LQ problems and some linear matrix inequality (LMI) conditions are established. Then, based on the equivalence between the solvability of the generalized difference Riccati equation (GDRE) and the LMI condition, it is proven that the solvability of the GDRE is sufficient and necessary for the well-posedness of the LQ problem. Moreover, the solvability of the GDRE and the feasibility of the LMI condition are established, and it is proven that the LQ problem is attainable through a certain feedback control when any of the four conditions is satisfied, and the optimal feedback control of the LQ problem is given using the properties of homogeneous Markov processes and the smoothness of the conditional expectation. Finally, a practical example is used to illustrate the validity of the theory.