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 


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  1. [1].
    ASPLUND, E. The potential of projections in Hilbert space (quoted in Edelstein [1]).Google Scholar
  2. [2].
    ASPLUND, E. Farthest point of sets in reflexive locally uniformly rotund Banach space. Israel J. of Maths 4 (1966) p 213–216.Google Scholar
  3. [3].
    ASPLUND, E. Frechet differentiability of convex functions. Acta Math 421 (1968) p 31–47.Google Scholar
  4. [4].
    ASPLUND, E. Topics in the theory of convex functions. Proceedings of NATO, Venice, june 1968. Aldo Ghizetti editor, Edizioni Oderisi.Google Scholar
  5. [1].
    BARANGER, J. Existence de solution pour des problèmes d'optimisation non convexe.Google Scholar
  6. [1]b.
    BARANGER, J. C.R.A.S. t 274 p 307.Google Scholar
  7. [2].
    BARANGER, J. Quelques résultats en optimisation non convexe.Google Scholar
  8. [2]b.
    BARANGER, J. Deuxième partie: Théorèmes d'existence en densité et application au contrôle. Thèse Grenoble 1973.Google Scholar
  9. [1].
    BARANGER, J., and TEMAM, R. Non convex optimization problems depending on a parameter. A paraitre.Google Scholar
  10. [1].
    EDELSTEIN, M. Farthest points of sets in uniformly convex Banach spaces. Israel J. of Math 4 (1966) p 171–176.Google Scholar
  11. [1].
    SMULIAN, V.L. Sur la dérivabilité de la norme dans l'espace de Banach. Dokl. Akad. Nauk SSSR (N.S) 27 (1940), p 643–648.Google Scholar
  12. [1].
    ZISLER, V. On some extremal problems in Banach spaces.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1973

Authors and Affiliations

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

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