Optimal Control of Viscous Fatigue Damage Models for Brittle Materials: Optimality Systems

• Applicant: Dr. Livia Betz
• Subject Area: Mathematics
• Term: since 2022
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 495294495

The proposed project aims at establishing first-order necessary optimality conditions in form of qualified optimality systems for a control problem (P) governed by a viscous fatigue damage model for brittle materials. This describes the evolution of the displacement and damage (state variables) under the influence of a time-dependent load (control) in a purely elastic brittle material. The mathematical model (state equation) consists of a non-linear equation coupled with a viscous history-dependent evolutionary variational inequality. The latter involves a history-variable which accounts for the fatigue experienced by the material.The problem has a non-linear, and more importantly, a non-smooth character. Thus, we are concerned with non-smooth optimization problems in function spaces. This excludes the application of standard adjoint techniques for the derivation of first-order necessary conditions in form of optimality systems. We emphasize that we intend to deal with the non-smooth structure of (P) by following a direct approach, i.e., we will not use classical regularization techniques that smoothen the problem. Moreover, (P) features control constraints as well as a non-smooth term in the objective.To achieve our main goal, we plan to address an auxiliary optimal control problem (Q). This differs from (P) only regarding its state system, which this time consists of a suitable approximation of the state equation in terms of a non-smooth coupled PDE system. One could even dispose of two different ways to approximate the viscous fatigue damage model, which allows us to present an alternative to our optimization problem (Q).In order to be able to derive optimality systems for (P), we shall perform a careful investigation in more steps, so that the goals we wish to accomplish in the proposed project are:1. To perform a rigorous analysis of the viscous fatigue damage model with regard to existence and uniqueness of solutions, which shall allow us to define the underlying control-to-state map.2. To establish necessary optimality conditions of strong stationary type for the control problem (Q) and its aforementioned alternative.3. To derive optimality systems for the main optimization problem (P). Here we aim to pass to the limit in the strong stationary optimality systems associated to the two alternative approximations of (P). This shall lead to two alternative optimality systems, which may complement each other, and eventually yield a more accurate optimality system for (P).

• DFG Programme: Research Grants