Norm perturbation of supremum problems

  • J. Baranger
Optimal Control
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3)


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.


Banach Space Normed Linear Space Norm Topology Reflexive Banach Space Weakly Compact 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

  • J. Baranger
    • 1
  1. 1.Institut de Mathématiques AppliquéesGrenoble CédexFrance

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