A phase field approach for optimal boundary control of damage processes in two-dimensional viscoelastic media

TL;DR

This paper develops a phase field model for damage in 2D viscoelastic media, establishing well-posedness and analyzing an optimal boundary control problem to minimize damage deviations.

Contribution

It introduces a novel phase field damage model with boundary control in viscoelastic media, proving existence, uniqueness, and optimal control solutions.

Findings

Proved global existence and uniqueness of solutions.

Established existence of optimal boundary controls.

Analyzed approximations for the control problem.

Abstract

In this work we investigate a phase field model for damage processes in two-dimensional viscoelastic media with nonhomogeneous Neumann data describing external boundary forces. In the first part we establish global-in-time existence, uniqueness, a priori estimates and continuous dependence of strong solutions on the data. The main difficulty is caused by the irreversibility of the phase field variable which results in a constrained PDE system. In the last part we consider an optimal control problem where a cost functional penalizes maximal deviations from prescribed damage profiles. The goal is to minimize the cost functional with respect to exterior forces acting on the boundary which play the role of the control variable in the considered model. To this end, we prove existence of minimizers and study a family of "local" approximations via adapted cost functionals.