Optimal control of an abstract evolution variational inequality with application to homogenized plasticity

TL;DR

This paper develops a regularization approach for optimal control problems governed by abstract evolution variational inequalities involving maximal monotone operators, with applications to homogenized elastoplasticity.

Contribution

It introduces a regularization method using Yosida approximation for non-Gâteaux differentiable state equations and derives optimality conditions, applying the theory to elastoplasticity.

Findings

Convergence of regularized solutions to the original problem.

Necessary and sufficient optimality conditions established.

Application to homogenized quasi-static elastoplasticity demonstrated.

Abstract

The paper is concerned with an optimal control problem governed by a state equation in form of a generalized abstract operator differential equation involving a maximal monotone operator. The state equation is uniquely solvable, but the associated solution operator is in general not G\^ateaux-differentiable. In order to derive optimality conditions, we therefore regularize the state equation and its solution operator, respectively, by means of a (smoothed) Yosida approximation. We show convergence of global minimizers for regularization parameter tending to zero and derive necessary and sufficient optimality conditions for the regularized problems. The paper ends with an application of the abstract theory to optimal control of homogenized quasi-static elastoplasticity.