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

TL;DR

This paper introduces a new class of sweeping process control problems with time delay, establishing their well-posedness and stability of discrete approximations, thus enabling effective numerical solutions.

Contribution

It is the first to analyze well-posedness and stability of discrete approximations for time-delayed sweeping control systems, bridging continuous and discrete optimization methods.

Findings

Established existence and uniqueness of solutions.

Constructed stable discrete approximations.

Validated convergence of solutions.

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.