Optimal control of a perturbed sweeping process via discrete discrete approximations

TL;DR

This paper develops a discrete approximation method for solving an optimal control problem involving a perturbed sweeping process with irregular data, providing necessary optimality conditions and convergence results.

Contribution

It introduces a novel discrete approximation approach for a complex sweeping process control problem with irregular data, enabling effective solution derivation.

Findings

Discrete approximations strongly converge to the original problem's solution.

Derived necessary optimality conditions using second-order variational analysis.

Provided a framework for handling irregular data in sweeping process control.

Abstract

The paper addresses an optimal control problem for a perturbed sweeping process of the rate-independent hysteresis type described by a controlled "play and stop" operator with separately controlled perturbations. This problem can be reduced to dynamic optimization of a state-constrained unbounded differential inclusion with highly irregular data that cannot be treated by means of known results in optimal control theory for differential inclusions. We develop the method of discrete approximations, which allows us to adequately replace the original optimal control problem by a sequence of well-posed finite-dimensional optimization problems whose optimal solutions strongly converge to that of the controlled perturbed sweeping process. To solve the discretized control systems, we derive effective necessary optimality conditions by using second-order generalized differential tools of variational analysis that explicitly calculated in terms of the given problem data.