On the no-gap second-order optimality conditions for a non-smooth semilinear elliptic optimal control

TL;DR

This paper develops a comprehensive second-order optimality theory for non-smooth semilinear elliptic control problems involving max-functions, addressing the challenge of non-Gâteaux differentiability through structural assumptions and curvature functionals.

Contribution

It introduces a no-gap second-order optimality condition framework for non-smooth PDE control problems, with precise characterizations of differentiability and active set structures.

Findings

Established necessary and sufficient second-order conditions

Characterized Gâteaux-differentiability via adjoint state on active sets

Developed a curvature functional approach for no-gap conditions

Abstract

This work is concerned with second-order necessary and sufficient optimality conditions for optimal control of a non-smooth semilinear elliptic partial differential equation, where the nonlinearity is the non-smooth max-function and thus the associated control-to-state operator is in general not G\^{a}teaux-differentiable. In addition to standing assumptions, two main hypotheses are imposed. The first one is the G\^{a}teaux-differentiability at the considered control of the objective functional and it is precisely characterized by the vanishing of an adjoint state on the active set. The second one is a structural assumption on the 'almost' active sets, i.e., the sets of all points at which the values of the interested state are 'close' to the non-differentiability point of the max-function. We then derive a 'no-gap' theory of second-order optimality conditions in terms of an abstract curvature functional, i.e., for which the only change between necessary or sufficient second-order optimality conditions is between a strict and non strict inequality.