Optimal control for stochastic discrete-time systems with multiple input-delays

The main purpose of the paper is to settle the stochastic linear quadratic control problem for systems with multiple input-delays (SDLQ) which is very intractable and remains to be solved. We introduce a different version of stochastic discrete-time maximum principle (SDMP) where it is shown that the auxiliary variable depends on the optimal system state through a stochastic matrix and the expectation of the relationship matrix happens to be the solution to the standard generalized Riccati equation with the same dimension as the origin system. The relationship explores the key difference of stochastic LQ from the deterministic one. It enables us to obtain the kernel of the optimal cost function for stochastic control and further the analytical and explicit solution to the stochastic LQ control problem with multiple input-delays.