Sequence-based necessary second-order optimality conditions for semilinear elliptic optimal control problems with nonsmooth data

Abstract

The main goal of this note is to formulate sequence-based necessary second-order optimality conditions for a semilinear elliptic optimal control problem, with a pointwise pure state constraint and a pointwise mixed state-control constraint, governed by nonsmooth functions. Using a modification of the Dubovitskii–Milyutin scheme, we develop necessary second-order conditions for optimality, which depend on given sequences, in terms of sequential tangent sets and sequential directional derivatives for not necessarily twice differentiable mappings.