Norm perturbation of supremum problems

Abstract

Let E be a normed linear space, S a closed bounded subset of E and J an u.s.c. (for the norm topology) and bounded above mapping of S into ℝ.

It is well known that in general there exists no s ∈ S such that

$$J\left( {\bar s} \right) = \mathop {Sup}\limits_{s \in S} J\left( s \right)$$

(even if S is weakly compact).

For J(s) = ∥x−s∥ (with x given in E), Edelstein, Asplund and Zisler have shown, under various hypotheses on E and S, that the set

$$\left( s \right) = \{ x \in E\left| \exists \right. \bar s \in S such that \left\| {\bar s - x} \right\| = \mathop {Sup}\limits_{x \in S} \left\| {s - x} \right\|\}$$

is dense in E.

Here we give analogous results for the problem

$$\mathop {Sup}\limits_{s \in S} (J(s) + \left\| {s - x} \right\|)$$

These results generalize those of Asplund and Zisler and allow us to obtain existence theorems for perturbed problems in optimal control.

This work is part of a thesis submitted at Université de Grenoble in 1973.