An Optimal Control Problem Governed by a Regularized Phase-Field Fracture Propagation Model

This paper is concerned with an optimal control problem governed by a fracture model using a phase-field technique. To avoid the non-differentiability due to the irreversibility constraint on the fracture growth, the phase-field fracture model is relaxed using a penalization approach. Existence of a solution to the penalized fracture model is shown and existence of at least one solution for the regularized optimal control problem is established. Moreover, the linearized fracture model is considered and used to establish first order necessary conditions as well as to discuss QP-approximations to the nonlinear optimization problem. A numerical example suggests that these can be used to obtain a fast convergent algorithm.