Second-order conditions for bang-bang control of elliptic equations in arbitrary dimensions

TL;DR

This paper establishes second-order optimality conditions for bang-bang controls in elliptic PDEs across any spatial dimension, using Bessel potential space theory to characterize quadratic growth of the objective.

Contribution

It introduces a novel proof method that applies in arbitrary dimensions, extending previous results limited to specific cases.

Findings

Second-order conditions characterize quadratic growth in bang-bang control problems.

The method applies to elliptic PDEs in any spatial dimension.

The approach broadens the applicability of optimality conditions in control theory.

Abstract

We consider an optimal control problem governed by a semilinear PDE in cases where the optimal control is of bang-bang type. By utilizing the theory of Bessel potential space, we characterize quadratic growth of the objective via a second-order optimality condition. In contrast to previous contributions, our method of proof works in arbitrary spatial dimensions.