Approximation of shape optimization problems with non-smooth PDE constraints

TL;DR

This paper develops a method to approximate shape optimization problems governed by non-smooth PDEs, enabling analysis and computation despite non-differentiability and non-convexity issues.

Contribution

It introduces a variational control framework for non-smooth PDE-constrained shape optimization, including density results and convex approximations for optimality analysis.

Findings

Established density property for control set

Derived strong stationary optimality conditions

Linked approximations to non-smooth PDE shape optimization

Abstract

This paper is concerned with a shape optimization problem governed by a non-smooth PDE, i.e., the nonlinearity in the state equation is not necessarily differentiable. We follow the functional variational approach of [40] where the set of admissible shapes is parametrized by a large class of continuous mappings. This methodology allows for both boundary and topological variations. It has the advantage that one can rewrite the shape optimization problem as a control problem in a function space. To overcome the lack of convexity of the set of admissible controls, we provide an essential density property. This permits us to show that each parametrization associated to the optimal shape is the limit of global optima of non-smooth distributed optimal control problems. The admissible set of the approximating minimization problems is a convex subset of a Hilbert space of functions. Moreover, its structure is such that one can derive strong stationary optimality conditions [6]. The present manuscript provides the basis for the investigations from [5], where necessary conditions in form of an optimality system have been recently established.