Well-posedness and Stability of Discrete Approximations for Controlled Sweeping Processes with Time Delay

Abstract: This paper addresses, for the first time in the literature, optimal control problems for dynamic systems governed by a novel class of sweeping processes with time delay. We establish well-posedness of such processes, in the sense of the existence and uniqueness of feasible trajectories corresponding to feasible controls under fairly unrestrictive assumptions. Then we construct a well-posed family of discrete approximations and find efficient conditions under the discretized time-delayed sweeping process exhibits stability with respect to strong convergence of feasible and optimal solutions. This creates a bridge between optimization of continuous-time and discrete-time sweeping control systems and justifies the effective use of discrete approximations in deriving optimality conditions and numerical techniques to solve the original time-delayed sweeping control problems via discrete approximations.