An efficient recursive shooting method for the optimal control of time-varying systems with state time-delay

This paper presents a recursive shooting method for solving the optimal control problem of linear time-varying systems with state time-delay. In this approach, the original time-delay optimal control problem is first transformed into a sequence of linear two-point boundary value problems (TPBVPs) without delay and advance terms. Then, by using a shooting method for the solution of latter sequence in a recursive manner, an optimal control law is achieved which consists feedback and forward terms. The feedback term is determined by solving a matrix Riccati differential equation. The forward term is an infinite sum of adjoint vectors, which can be obtained by solving the above-mentioned sequence of linear non-delay TPBVPs. The convergence analysis of the proposed approach is also provided. Finally, some comparative results are included to illustrate the effectiveness of the proposed approach.