Optimal control for stochastic systems with multiple input delays

In this paper, the stochastic LQR problem for discrete-time systems with multiplicative noise and multiple input delays is considered. Specifically, there are two input channels, u and w, among which u is delay-free and w is with a delay d. Let the augmented state include the state and a finite window history of length d of w. The problem is solved by virtue of the maximum principle for stochastic systems. Step by step, we establish the linear homogeneous relation between the costate at k and the augmented state at k + 1. The associated coefficients satisfy d + 1 partial Riccati equations of the same order of the plant. The optimal controllers, which both own the form of linear functions of the augmented state, can be calculated via the solution of the aforementioned Riccati equations.