Stabilization and Optimal Control for Discrete-Time Markov Jump Linear System With Multiplicative Noises and Input Delays: A Complete Solution

This article investigates the stabilization and optimal control problems for Markov jump linear system (MJLS) with multiplicative noises and input delays. By overcoming the substantive difficulty resulting from the invalidity of the separation principle, we provide a complete solution to the addressed problems by means of 1) necessary and sufficient solvability, and the analytical formula on optimal finite horizon control in line with a generalized coupled difference Riccati equation; and 2) necessary and sufficient stabilizability, and the explicit expression of the optimal controller on infinite horizon according to a delayed generalized coupled algebraic Riccati equation (D-GCARE). It is shown that the MJLS with multiplicative noises and input delays is mean square stabilizable if and only if the D-GCARE has a specific solution. Our main results are attained through the creation of a novel delayed stochastic Markov maximum principle as well as the construction of a novel class of delayed Markov Lyapunov function.