Optimal Control for Markov Jump Linear System with Multiplicative Noise and Input Delay

This paper investigates the finite-horizon quadratic optimal control problem of discrete-time Markovian jump linear systems subject to multiplicative noises and input delays. A new stochastic Markovian jump maximum principle is developed for dealing this control problem. Applying this principle to the case of stochastic jumping parameter systems with one-step input delay, a necessary and sufficient condition for the problem admitting a unique solution is given. Under this condition, an explicit analytical optimal controller and the corresponding optimal cost are presented in terms of two generalized jumping coupled difference equations with the same dimension as the original state. The key technique is to establish relations between the optimal costate and the state.