Necessary second-order optimality conditions in distributed-boundary semilinear elliptic optimal control with twice directionally differentiable functions

Abstract

In this paper we investigate second-order optimality conditions for distributed-boundary control problems governed by semilinear elliptic systems, where the objective function may not be of class C2 , in the presence of pointwise pure state constraints and pointwise mixed state-control constraints. Using local approximations and directional derivatives for twice directionally differentiable mappings, we examine necessary second-order optimality conditions for the problems, by virtue of a Lagrange multiplier theorem in mathematical programs.