First- and second-order optimality conditions in the sparse optimal control of Cahn--Hilliard systems

Abstract

This paper deals with the sparse distributed control of viscous and nonviscous Cahn--Hilliard systems. We report on results concerning first-order necessary and second-order sufficient optimality conditions that have recently established by the authors. The analysis covers both the cases when the nonlinear double well potential governing the evolution is of either regular or logarithmic type. A major difficulty originates from the sparsity-enhancing term in the cost functional which typically is nondifferentiable.