Second-order sufficient conditions for sparse optimal control of singular Allen–Cahn systems with dynamic boundary conditions

Jürgen Sprekels; Fredi Tröltzsch

doi:10.3934/dcdss.2023163

Article Contents

2023, Volume 16, Issue 12: 3784-3812. Doi: 10.3934/dcdss.2023163

This issue Previous Article Kobayashi-Warren-Carter system of singular type under dynamic boundary condition Next Article On fractional and classical hyperbolic obstacle-type problems

Second-order sufficient conditions for sparse optimal control of singular Allen–Cahn systems with dynamic boundary conditions

Jürgen Sprekels^1, , and
Fredi Tröltzsch^2,

1.
Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D-10117 Berlin, Germany
2.
Institut für Mathematik der Technischen Universität Berlin, Straße des 17. Juni 136, D-10623 Berlin, Germany

^*Corresponding author: Jürgen Sprekels
^*Corresponding author: Jürgen Sprekels

Dedicated to our dear friend Pierluigi Colli on the occasion of his 65th birthday

Received: March 2023

Early access: September 8, 2023

Published online: September 8, 2023

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Abstract

In this paper we study the optimal control of a parabolic initial-boundary value problem of Allen–Cahn type with dynamic boundary conditions. Phase field systems of this type govern the evolution of coupled diffusive phase transition processes with nonconserved order parameters that occur in a container and on its surface, respectively. It is assumed that the nonlinear functions driving the physical processes within the bulk and on the surface are double well potentials of logarithmic type whose derivatives become singular at the boundary of their respective domains of definition. For such systems, optimal control problems have been studied in the past. We focus here on the situation when the cost functional of the optimal control problem contains a nondifferentiable term like the $ L^1 $-norm leading to sparsity of optimal controls. For such cases, we derive second-order sufficient conditions for locally optimal controls.

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Mathematics Subject Classification: Primary: 49J50, 49J52; Secondary: 35K20, 35K55, 49K20.

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