On the optimal control of some nonsmooth distributed parameter systems arising in mechanics

TL;DR

This paper explores the analysis and optimal control of second-kind variational inequalities in mechanics, addressing their mathematical challenges, applications, and the existence of optimality conditions.

Contribution

It provides a comprehensive overview of the applications, analytical difficulties, and optimal control frameworks for nonsmooth variational inequalities of the second kind.

Findings

Discussion on the existence of Lagrange multipliers

Derivation of optimality systems for local minima

Identification of main analytical and numerical challenges

Abstract

Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of the second kind, with a twofold purpose. First, we aim at giving a glance at some of the most prominent applications of these types of variational inequalities in mechanics, and the related analytical and numerical difficulties. Second, we consider optimal control problems constrained by these variational inequalities and provide a thorough discussion on the existence of Lagrange multipliers and the different types of optimality systems that can be derived for the characterization of local minima. The article ends with a discussion of the main challenges and future perspectives of this important problem class.