Optimal Control of Semilinear Elliptic Partial Differential Equations with Non-Lipschitzian Nonlinearities

TL;DR

This paper investigates optimal control problems governed by semilinear elliptic PDEs with non-Lipschitz nonlinearities, establishing differentiability of the solution map and deriving first-order optimality conditions despite nondifferentiability in the operator.

Contribution

It demonstrates the Fréchet differentiability of the solution map for a class of PDEs with non-Lipschitz nonlinearities and derives first-order optimality conditions in this challenging setting.

Findings

Solution map is Fréchet differentiable despite non-Lipschitz nonlinearities.

First-order optimality conditions are established in non-Muckenhoupt weighted Sobolev spaces.

Raises questions on regularity, second-order conditions, and finite element discretizations.

Abstract

We study optimal control problems that are governed by semilinear elliptic partial differential equations that involve non-Lipschitzian nonlinearities. It is shown that, for a certain class of such PDEs, the solution map is Fr\'{e}chet differentiable even though the differential operator contains a nondifferentiable term. We exploit this effect to establish first-order necessary optimality conditions for minimizers of the considered control problems. The resulting KKT-conditions take the form of coupled PDE-systems that are posed in non-Muckenhoupt weighted Sobolev spaces and raise interesting questions regarding the regularity of optimal controls, the derivation of second-order optimality conditions, and the analysis of finite element discretizations.