Optimization of Fully Controlled Sweeping Processes

TL;DR

This paper develops necessary optimality conditions for controlled sweeping processes using advanced variational analysis, providing a systematic approach and demonstrating its effectiveness through illustrative examples.

Contribution

It introduces a novel method of deriving verifiable optimality conditions for sweeping processes via discrete approximations and second-order variational tools.

Findings

Derived verifiable necessary optimality conditions for sweeping processes.

Established a discrete approximation method for the control problem.

Validated the approach with three illustrative examples.

Abstract

The paper is devoted to deriving necessary optimality conditions in a general optimal control problem for dynamical systems governed by controlled sweeping processes with hard-constrained control actions entering both polyhedral moving sets and additive perturbations. By using the first-order and mainly second-order tools of variational analysis and generalized differentiation, we develop a well-posed method of discrete approximations, obtain optimality conditions for solutions to discrete-time control systems, and then establish by passing to the limit verifiable necessary optimality conditions for local minimizers of the original controlled sweeping process that are expressed entirely in terms of its given data. The efficiency of the obtained necessary optimality conditions for the sweeping dynamics is illustrated by solving three nontrivial examples of their own interest.