Optimal control for discrete-time singular systems with multiplicative-noise and input delay

This paper deals with the finite-horizon linear quadratic (LQ) optimal control problem for the discrete-time singular systems with multiplicative-noise and input delay. Firstly, by utilizing the incremental analysis method, the maximum principle is established for the singular stochastic system in discrete-time form. Then a sufficient criterion is given which guarantees not only the existence but also the uniqueness of the optimal controller. With such kind of controllers designed, the mathematical expectation of the cost function reaches its minimum value which is also given in explicit form. The criteria obtained in this paper are presented in the recursive equality form, which can be verified easily by effective algorithms. A numerical example is provided in the end of the paper which demonstrates the effectiveness of the obtained theoretical results.