Optimal control of a non-smooth quasilinear elliptic equation

TL;DR

This paper develops optimal control methods for a non-smooth quasilinear elliptic PDE, deriving stationarity conditions and a numerical scheme using regularization and semi-smooth Newton methods.

Contribution

It introduces a regularization approach for non-smooth PDE control problems, establishing stationarity conditions and an implementable optimality system.

Findings

Derived C- and strong stationarity conditions.

Proved equivalence of stationarity conditions under certain assumptions.

Developed a semi-smooth Newton method for numerical solution.

Abstract

This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not G\^ateaux differentiable. This leads to a control-to-state operator that is directionally but not G\^ateaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a PC^1 function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.