Optimal control of a quasilinear parabolic equation and its time discretization

TL;DR

This paper studies the optimal control of a quasilinear parabolic equation, focusing on existence, discretization, and convergence of controls, with applications to models like the anisotropic Allen-Cahn equation for crystal growth.

Contribution

It establishes the existence and Lipschitz continuity of the control-to-state operator for both continuous and discretized problems, and proves convergence of discrete controls to continuous minimizers.

Findings

Existence of control-to-state operator for discretized and continuous problems

Lipschitz continuity of the control-to-state operator

Convergence of discrete optimal controls to continuous minimizers

Abstract

In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on the kind of problems arising for example when controlling the anisotropic Allen-Cahn equation as a model for crystal growth. Motivated by this application we consider the state equation as a result of a gradient flow of an energy functional. The quasilinear term is strongly monotone and obeys a certain growth condition and the lower order term is non-monotone. The state equation is discretized implicitly in time with piecewise constant functions. The existence of the control-to-state operator and its Lipschitz-continuity is shown for the time discretized as well as for the time continuous problem. Latter is based on the convergence proof of the discretized solutions. Finally we present for both the existence of global minimizers. Also convergence of a subsequence of time discrete optimal controls to a global minimizer of the time continuous problem can be shown. Our results hold in arbitrary space dimensions.