Optimal Control of a Quasi-Variational Sweeping Process

TL;DR

This paper develops a framework for optimal control of complex parabolic quasi-variational inequalities related to material growth, introducing regularization, approximation, and deriving necessary optimality conditions using advanced variational analysis.

Contribution

It formulates a novel optimal control problem for quasi-variational inequalities and establishes existence and optimality conditions using advanced mathematical tools.

Findings

Existence of optimal solutions for the control problem.

Development of regularization and approximation procedures.

Derivation of necessary optimality conditions using variational analysis.

Abstract

The paper addresses the study of a class of evolutionary quasi-variational inequalities of the parabolic type arising in the formation and growth models of granular and cohensionless materials. Such models and their mathematical descriptions are highly challenging and require powerful tools of their analysis and implementation. We formulate a space-time continuous optimal control problem for a basic model of this type, develop several regularization and approximation procedures, and establish the existence of optimal solutions for the time-continuous and space-discrete problem. Viewing a version of this problem as a controlled quasi-variational sweeping process leads us to deriving necessary optimality conditions for the fully discrete problem by using the advanced machinery of variational analysis and generalized differentiation.