Optimal control of a non-smooth elliptic PDE with non-linear term acting on the control

TL;DR

This paper derives first-order optimality conditions for a control-constrained optimization problem governed by a non-smooth elliptic PDE with a control-dependent non-linearity, extending the theoretical framework for such problems.

Contribution

It establishes strong stationary optimality conditions for non-smooth elliptic PDE control problems, which were previously difficult due to non-differentiability.

Findings

Derived conditions for strong stationarity in non-smooth PDE control problems

Extended the theoretical understanding of non-smooth optimality systems

Applied findings to non-smooth shape optimization in recent work

Abstract

This paper continues the investigations from [7] and is concerned with the derivation of first-order conditions for a control constrained optimization problem governed by a non-smooth elliptic PDE. The control enters the state equation not only linearly but also as the argument of a regularization of the Heaviside function. The non-linearity which acts on the state is locally Lipschitz-continuous and not necessarily differentiable, i.e., non-smooth. This excludes the application of standard adjoint calculus. We derive conditions under which a strong stationary optimality system can be established, i.e., a system that is equivalent to the purely primal optimality condition saying that the directional derivative of the reduced objective in feasible directions is nonnegative. For this, two assumptions are made on the unknown optimizer. Some of the presented findings are employed in the recent contribution [8], where limit optimality systems for non-smooth shape optimization problems [7] are established.