Analysis of a space-time phase-field fracture complementarity model and its optimal control formulation

TL;DR

This paper develops optimality conditions for a phase-field fracture model formulated as a nonlinear optimization problem, addressing regularity issues and deriving necessary and sufficient conditions for optimal control.

Contribution

It introduces a novel framework for deriving optimality conditions for phase-field fracture models with a focus on regularity and control constraints.

Findings

Derived necessary optimality conditions for the model.

Established regularity conditions for the control problem.

Formulated a tracking-type optimal control problem with constraints.

Abstract

The purpose of this work is the formulation of optimality conditions for phase-field optimal control problems. The forward problem is first stated as an abstract nonlinear optimization problem, and then the necessary optimality conditions are derived. The sufficient optimality conditions are also examined. The choice of suitable function spaces to ensure the regularity of the nonlinear optimization problem is a true challenge here. Afterwards the optimal control problem with a tracking type cost functional is formulated. The constraints are given by the previously derived first order optimality conditions of the forward problem. Herein regularity is proven under certain conditions and first order optimality conditions are formulated.