Tracking Problem for Itô Stochastic System with Input Delay

In this article, linear quadratic tracking (LQT) problem for Itô stochastic system with time delay in control input is studied. The difference between LQT and linear quadratic regulation (LQR) is that the state and control input are desired to track a reference trajectory. It is noteworthy that the separation principle is not valid for Itô stochastic system. In this paper, we design an optimal controller based on the maximum principle, in which the optimal controller minimizes the linear quadratic performance index of state and control input tracking errors. Based on the coupled Riccati-like equation, the explicit expression of the optimal controller is given. The key technology of this paper is to solve forward and backward stochastic differential equations (FBSDEs). Finally, a numerical example is provided to validate the results.