Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Nemitsky's operators and lower closure theorems

Abstract

This paper focuses on certain analytic criteria given by the authors in earlier works, for the geometric property of upper semicontinuity of set-valued functions, used in the proofs of lower closure theorems, and hence in existence theory. In particular, it is observed that, under Filippov-type condition (namely, when the set of controls is bounded in measure or in norm), mere Carathéodory-type continuity of the relevant functionsf is sufficient to guarantee a weak form of property (Q), and in turn the lower closure theorems.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Filippov, A. F.,On Certain Questions in the Theory of Optimal Control, SIAM Journal on Control, Vol. 1, pp. 76–84, 1962.

  2. 2.

    Cesari, L.,Existence Theorems for Weak and Usual Optimal Solutions in Lagrange Problems with Unilateral Constraints, I and II, Transactions of the American Mathematical Society, Vol. 124, pp. 369–412 and pp. 413–429, 1966.

  3. 3.

    Cesari, L.,Closure Theorems for Orientor Fields, Bulletin of the American Mathematical Society, Vol. 79, pp. 684–689, 1973.

  4. 4.

    Cesari, L.,Closure Theorems for Orientor Fields and Weak Convergence, Archive for Rational Mechanics and Analysis, Vol. 55, pp. 332–356, 1974.

  5. 5.

    Cesari, L.,Lower Semicontinuity and Lower Closure Theorems without Seminormality Conditions, Annali di Matematica Pura e Applicata, Vol. 98, pp. 381–397, 1974.

  6. 6.

    Cesari, L.,A Necessary and Sufficient Condition for Lower Semicontinuity, Bulletin of the American Mathematical Society, Vol. 80, pp. 467–472, 1974.

  7. 7.

    Cesari, L., andSuryanarayana, M. B.,Closure Theorems Without Seminormality Conditions, Journal of Optimization Theory and Applications, Vol. 15, pp. 441–465, 1975.

  8. 8.

    Cesari, L., andSuryanarayana, M. B.,Convexity and Property (Q) in Optimal Control Theory, SIAM Journal on Control, Vol. 12, pp. 705–720, 1974.

  9. 9.

    Olech, C.,Existence Theorems for Optimal Problems with Vector Valued Cost Functions, Transactions of the American Mathematical Society, Vol. 136, pp. 157–180, 1969.

  10. 10.

    Lasota, A., andOlech, C.,On Cesari's Semicontinuity Condition for Set Valued Mappings, Bulletin de l'Academie Polonaise des Sciences, Serie des Sciences Mathematiques, Astronomiques et Physiques, Vol. 16, pp. 711–716, 1968.

  11. 11.

    Bidaut, M. F.,Quelques Resultats d'Existence pour des Problemes de Controle Optimal, Comptes Rendus Hebdomadaires des Seances de l'Academie des Sciences, Paris, Vol. 274, pp. 62–65, 1972.

  12. 12.

    Berkovitz, L. D.,Existence and Lower Closure Theorems for Abstract Control Problems, SIAM Journal on Control, Vol. 12, pp. 27–42, 1974.

  13. 13.

    Berkovitz, L. D.,Lower Closure and Existence Theorems in Optimal Control, Proceedings of the International Conference on Differential Equations, Edited by H. A. Antosiewicz, Academic Press, New York, New York, 1975.

  14. 14.

    Berkovitz, L. D.,A Lower Closure Theorem for Abstract Control Problems with L p-Bounded Controls, Journal of Optimization Theory and Applications, Vol. 14, pp. 521–528, 1974.

  15. 15.

    Suryanarayana, M. B.,Remarks on Lower Semicontinuity and Lower Closure, Journal of Optimization Theory and Applications, Vol. 19, pp. 125–140, 1976.

  16. 16.

    Krasnosel'skii, M. A., et al.,Integral Operators in Spaces of Summable Functions (in Russian), Nauka, Moscow, USSR, 1966.

  17. 17.

    Cesari, L.,Problems of Optimization, I and II, Springer-Verlag, New York, New York (to appear).

Download references

Author information

Additional information

This work has been partially supported by Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cesari, L., Suryanarayana, M.B. Nemitsky's operators and lower closure theorems. J Optim Theory Appl 19, 165–183 (1976). https://doi.org/10.1007/BF00934059

Download citation

Key Words

  • Lower closure theorems
  • Nemitsky operators
  • upper semicontinuity
  • control theory
  • existence theorems
  • orientor fields
  • functional analysis