Optimal Control Governed by a Regularized Fracture Propagation Model: Optimality Conditions and Numerical Methods

Abstract

This thesis addresses the analysis of an optimal control problem in the context of fracture or damage propagation. The problem formulation is given by a tracking type optimal control problem which is governed by a state equation being a (system of) internally coupled regularized quasilinear elliptic partial differential equation(s), which is derived from a variational phase-field fracture propagation model. Due to the coupled structure and the involved need for improved regularity results for the state equation, well-posedness, uniqueness, as well as differentiability and Lipschitz continuity results of the associated solution operator are both interesting and challenging. As a consequence of the state equation being nonlinear, the objective functional of the overall optimal control problem is further nonconvex. <br /> After a brief introduction and a general overview of the fracture propagation model problem, we recapitulate the derivation and motivation of the model optimal control problem from several key publications of the literature. The main results of this thesis are contributions towards the topic of optimal control theory: We study necessary and sufficient optimality conditions of second-order, which do not involve a two-norm discrepancy, and have a minimal gap. Here, we extend recent publications concerning solution and first-order condition results, which cannot deliver sufficient conditions for the optimal control problem due to the objective functional being nonconvex. In the context of first-order necessary conditions and for a setting restricted to one time step, we prove convergence of the dual variables in the penalization (w.r.t. the fracture irreversibility) limit. We can then ensure that the limits of both the primal and dual variables suffice to a stationarity condition that is associated to the related unpenalized optimal control problem. Note that since we recover the to the nonhealing presumption associated inequality constraints in the limit, the obtained stationarity condition is the optimality condition of a mathematical program with complementarity constraints (MPCC), i.e. our work is also related to this field. The final main contribution of this thesis is the study of local quadratic convergence of the sequential quadratic programming (SQP) method for the fracture propagation optimal control problem. Here, we assume rather weak (sigma-strongly active) second-order sufficient conditions, which are close to the minimal-gap second-order conditions, both studied in previous chapters. However, this means that for the theoretical proofs we have to confine the involved SQP-subproblem to certain local neighborhoods. We prove local quadratic convergence of the SQP method both for subproblems localized in L-infinity and L-2. Finally, the theoretical convergence result are substantiated by numerical simulations of the SQP method for a simplified phase-field optimal control problem.