Second-order optimality conditions for an optimal control problem governed by a regularized phase-field fracture propagation model

Abstract

We prove second-order optimality conditions for an optimal control problem of tracking-type for a time-discrete regularized phase-field fracture or damage propagation model. The energy minimization problem describing the fracture process contains a penalization term for violation of the irreversibility condition in the fracture growth process, as well as a viscous regularization corresponding to a time-step restriction in a temporal discretization of the problem. In the control problem, the energy minimization problem is replaced by its Euler–Lagrange equations. While the energy minimization functional is convex due to the viscous approximation, the associated Euler–Lagrange equations are of quasilinear type, making the control problem nonconvex. We prove second-order necessary as well as second-order sufficient optimality conditions without two-norm discrepancy.

Keywords:

• Optimal control
• phase-field
• regularized fracture model
• second-order necessary conditions
• second-order sufficient conditions

AMS CLASSIFICATIONS:

• 49K20
• 74R10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer NE 1941/1-1 - SPP 1962.