Full Stability for a Class of Control Problems of Semilinear Elliptic Partial Differential Equations

TL;DR

This paper characterizes full Lipschitzian and H"olderian stability for control problems governed by semilinear elliptic PDEs, showing their equivalence under certain conditions and providing explicit criteria.

Contribution

It provides explicit characterizations of full stability for a class of control problems with perturbed cost, state, and control sets, and proves the equivalence of Lipschitzian and H"olderian stability.

Findings

Full stability characterized explicitly for the control problems.

Lipschitzian and H"olderian stability are equivalent in this class.

The two stability properties are always equivalent when the control set is fixed, convex, and closed.

Abstract

We investigate full Lipschitzian and full H\"olderian stability for a class of control problems governed by semilinear elliptic partial differential equations, where all the cost functional, the state equation, and the admissible control set of the control problems undergo perturbations. We establish explicit characterizations of both Lipschitzian and H\"olderian full stability for the class of control problems. We show that for this class of control problems the two full stability properties are equivalent. In particular, the two properties are always equivalent in general when the admissible control set is an arbitrary fixed nonempty, closed, and convex set.