Analysis and optimal control of a damage model with penalty

Abstract

A viscous damage model including two damage variables - a local and a nonlocal one - coupled through a penalty term is investigated on three different levels: unique solvability, behaviour as the penalization parameter approaches ∞ and optimal control. Existence, uniqueness and regularity of the solutions are proven. In particular, we give an improved result regarding spacial regularity of the nonlocal damage. Lipschitz continuity as well as Fréchet-differentiability of the solution operators are established. We also analyse the behaviour for penalty parameter tending to ∞ of the considered damage model. It turns out that in the limit both damage variables coincide and satisfy a classical viscous damage model. Moreover, we find L∞ bounds for the penalized damage variables and their limit. Further, an optimal control problem governed by the damage model with penalty is considered, where the applied force is used as control. In this context, we derive necessary optimality conditions for a local optimum. As the associated control-to-state operator is not Gâteaux differentiable, standard adjoint calculus cannot be employed for deriving an optimality system. This was however possible under the strict complementarity assumption.