Stability for Bang-Bang Control Problems of Partial Differential Equations

TL;DR

This paper studies the stability of bang-bang control solutions for PDEs, establishing optimality conditions and criteria for solution stability under perturbations, with a focus on H"older stability in L^1.

Contribution

It provides new sufficient optimality conditions and stability criteria specifically for bang-bang control problems of PDEs without quadratic control costs.

Findings

Established a sufficient optimality condition for bang-bang controls.

Derived criteria for solution stability under linear perturbations.

Proved H"older stability of optimal controls in L^1.

Abstract

In this paper, we investigate solution stability for control problems of partial differential equations with the cost functional not involving the usual quadratic term for the control. We first establish a sufficient optimality condition for the optimal control problems with bang-bang controls. Then we obtain criteria for solution stability for the optimal control problems of bang-bang controls under linear perturbations. We prove H\"older stability of optimal controls in $L^1$.