Analysis and approximations of an optimal control problem for the Allen-Cahn equation

TL;DR

This paper analyzes and approximates an optimal control problem for the Allen-Cahn equation, providing new a-priori estimates that are valid under low regularity and without relying on spectral estimates, applicable to discretizations respecting interface thickness.

Contribution

It introduces a novel approach to derive a-priori estimates for the Allen-Cahn control problem that do not depend on spectral estimates and are valid under low regularity conditions.

Findings

A-priori estimates with polynomial dependence on 1/ε

Error bounds for control and state approximations

Estimates valid under low regularity assumptions

Abstract

The scope of this paper is the analysis and approximation of an optimal control problem related to the Allen-Cahn equation. A tracking functional is minimized subject to the Allen-Cahn equation using distributed controls that satisfy point-wise control constraints. First and second order necessary and sufficient conditions are proved. The lowest order discontinuous Galerkin - in time - scheme is considered for the approximation of the control to state and adjoint state mappings. Under a suitable restriction on maximum size of the temporal and spatial discretization parameters $k$, $h$ respectively in terms of the parameter $\epsilon$ that describes the thickness of the interface layer, a-priori estimates are proved with constants depending polynomially upon $1/ \epsilon$. Unlike to previous works for the uncontrolled Allen-Cahn problem our approach does not rely on a construction of an approximation of the spectral estimate, and as a consequence our estimates are valid under low regularity assumptions imposed by the optimal control setting. These estimates are also valid in cases where the solution and its discrete approximation do not satisfy uniform space-time bounds independent of $\epsilon$. These estimates and a suitable localization technique, via the second order condition (see \cite{Arada-Casas-Troltzsch_2002,Casas-Mateos-Troltzsch_2005,Casas-Raymond_2006,Casas-Mateos-Raymond_2007}), allows to prove error estimates for the difference between local optimal controls and their discrete approximation as well as between the associated state and adjoint state variables and their discrete approximations