Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions
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 diffuse phase transition processes with nonconserved order parameters that occur in a container and on its surface, respectively. It is assumed that the nonlinear function 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.