Locally H\"{o}lder continuity of the solution map to a boundary control problem with finite mixed control-state constraints

TL;DR

This paper proves that the solution map for a boundary control problem governed by semilinear elliptic equations is locally Hölder continuous under certain optimality conditions, contributing to understanding the stability of such control systems.

Contribution

It establishes the local Hölder continuity of the solution map in the $L^ abla$-norm for boundary control problems with mixed constraints, under second-order optimality conditions.

Findings

Solution map is locally Hölder continuous in $L^ abla$-norm.

Continuity holds when second-order optimality conditions are satisfied.

Provides stability analysis for boundary control problems with mixed constraints.

Abstract

The local stability of the solution map to a parametric boundary control problem governed by semilinear elliptic equations with finite mixed pointwise constraints is considered in this paper. We prove that the solution map is locally H\"{o}lder continuous in $L^\infty-$norm of control variable when the strictly nonnegative second-order optimality conditions are satisfied for the unperturbed problem.