Stochastic control for discrete-time systems with input delay

In this paper, both the finite-horizon and infinite-horizon stochastic control problems for multiplicative-noise discrete-time systems with input delay are investigated. The novelty contributions of the paper are twofold. First, a sufficient and necessary unique existing condition for the optimal control is obtained. The optimal controller is the feedback of the conditional expectation of the state. The feedback gain is computed with the solution to Riccati-type difference equation. Second, a stabilizing controller which minimizes the infinite-horizon performance index is derived by showing the convergence of the finite-horizon controller. It is worth emphasizing that under the exact observability, the system is stabilizable in the mean-square sense if and only if the algebraic Riccati-type equation have unique solution such that the specified matrix equation is positive definite. The key techniques developed are the establishing of the nonhomogeneous relationship between the state and the costate and the introduction of the specified Lyapunov function, which originates from the results of the finite-horizon optimal control. Moreover, the optimal controller is actually the `Smith Predictor´ in time domain for discrete-time stochastic multiplicative-noise delay systems which is new to the best of our knowledge.