Skip to main content
SpringerLink
Log in

The role of tangent and normal cones in the viability theory of differential inclusions

  • Chapter
  • First Online:
Nonlinear Analysis and Optimization

Part of the book series: Mathematical Programming Studies ((MATHPROGRAMM,volume 30))

Abstract

This paper is a survey on the role played by tangent and normal cones for the existence of solutions to differential inclusions defined on a constrained set.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J.P. Aubin and A. Cellina, Differential inclusions (Springer-Verlag, Berlin-Heidelberg-New York, 1984).

    MATH  Google Scholar 

  2. J.P. Aubin, A. Cellina and J. Nohel, “Monotone trajectories of multivalued dynamical systems”, Annali di Matematica Pura e Applicata 115 (1977) 99–117.

    Article  MathSciNet  Google Scholar 

  3. J.P. Aubin and F. Clarke, “Monotone invariant solutions to differential inclusions”, Journal of the London Mathematical Society 16 (1977) 357–366.

    Article  MATH  MathSciNet  Google Scholar 

  4. J.M. Bony, “Principe du maximum, inégalité de Harnack et unicité du problème de Cauchy pour les opérateurs elliptiques dégénérés”, Annales de l’Institut Fourier 19 (1969) 277–304.

    MATH  MathSciNet  Google Scholar 

  5. G. Bouligand, Introduction à la géométrie infinitésimale directe (Gauthier-Villars, Paris, 1932).

    Google Scholar 

  6. A. Bressan, “On differential relations with lower semicontinuous right-hand side”, Journal of Differential Equations 37 (1980) 89–97.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Brézis, “On a characterization of flow invariant sets”, Communications of Pure and Applied Mathematics 23 (1970) 261–263.

    Article  MATH  Google Scholar 

  8. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (North-Holland, Amsterdam, 1973).

    MATH  Google Scholar 

  9. C. Castaing, “Equation différentielle multivoque avec contrainte sur l’état dans les espaces de Banach”, Séminaire d’analyse convexe, Université de Montpellier (Montpellier, 1978).

    Google Scholar 

  10. F. Clarke, “Generalized gradients and applications”, Transactions of the American Mathematical Society 205 (1975) 247–262.

    Article  MATH  MathSciNet  Google Scholar 

  11. B. Cornet, “Existence of slow solutions for a class of differential inclusions”, Journal of Mathematical Analysis and Applications 96 (1983) 130–147.

    Article  MATH  MathSciNet  Google Scholar 

  12. B. Cornet, “Cône tangent et cône normal à un ensemble”, Thèse d’Etat, Université Paris IX (Paris, 1981).

    Google Scholar 

  13. B. Cornet and G. Haddad, “Viability theorems for second order differential inclusions”, CORE discussion paper 8326, Université Catholique de Louvain (Louvain-la-Neuve, 1983).

    Google Scholar 

  14. M.G. Crandall, “A generalization of Peano’s existence theorem and flow invariance”, Proceedings of the American Mathematical Society 36 (1972) 151–155.

    Article  MATH  MathSciNet  Google Scholar 

  15. K. Deimling, Ordinary differential equations in Banach spaces, Lectures Notes in Mathematics 596 (Springer-Verlag, Berlin-Heidelberg-New York, 1977).

    MATH  Google Scholar 

  16. A.J. Dubovickii and A.A. Miljutin, “Extremum problems in presence of restrictions”, USSR Computational Mathematics and Mathematical Physics 5 (1965) 1–80.

    Article  MathSciNet  Google Scholar 

  17. S. Gautier, “Equations différentielles multivoques sur un fermé, Publications Mathématiques de Pau, Université de Pau (Pau, 1976).

    Google Scholar 

  18. P. Hartman, “On invariant sets and on a theorem of Wazewski”, Proceedings of the American Mathematical Society 32 (1972) 511–520.

    Article  MATH  MathSciNet  Google Scholar 

  19. G. Haddad, “Monotone trajectories of differential inclusions and functional differential inclusions with memory”, Israel Journal of Mathematics 39 (1981) 83–100.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. Henry, “An existence theorem for a class of differential equations with multivalued right-hand side”, Journal of Mathematical Analysis and Applications 41 (1973) 179–186.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Larrieu, “Invariance d’un fermé pour un champ de vecteurs de Carathedory”, Publications Mathématiques de Pau, Université de Pau (Pau, 1981).

    Google Scholar 

  22. C. Malivert, “Optimisation sur une variété anguleuse ou un fermé non convexe”, Thèse de 3e cycle, Université de Pau (Pau, 1977).

    Google Scholar 

  23. R.M. Martin, “Differential equations on closed subsets of a Banach space”, Transactions of the American Mathematical Society 179 (1973) 399–414.

    Article  MATH  MathSciNet  Google Scholar 

  24. M. Nagumo, “Uber die Lage der Intergralkurven gewöhnlicher Differential-gleichungen”, Proceedings of the Physical and Mathematical Society of Japan 24 (1942) 551–559.

    MATH  MathSciNet  Google Scholar 

  25. N. Pavel and C. Ursescu, “Flow-invariant sets for autonomous second order differential equations and applications in Mechanics”, Preprints series in Mathematics 18, University of Bucharest (Bucharest, 1980).

    Google Scholar 

  26. J.P. Penot, “A characterization of tangential regularity”, Nonlinear Analysis, Theory, Methods and Applications 6 (1981) 625–643.

    Article  MathSciNet  Google Scholar 

  27. R.M. Redhefler, “The theorems of Bony and Brézis on flow invariant sets”, American Mathematics Monthly 79 (1972) 790–797.

    Google Scholar 

  28. R.T. Rockafellar, Convex analysis (Princeton University Press, Princeton, N.J., 1970).

    MATH  Google Scholar 

  29. R.T. Rockafellar, “Clarke’s tangent cones and the boundaries of closed sets in ℝn”, Nonlinear Analysis 3 (1979) 145–154.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Yarom, “Invariant solutions to differential inclusions with nonconvex right-hand side”, Discussion paper 107, Universität Bonn (Bonn, 1982).

    Google Scholar 

  31. J.A. Yorke, “Invariance for ordinary differential equations”, Mathematical Systems Theory 1 (1967) 353–372.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté de Droit et des Sciences économiques, Université de Nice, Nice, France

    Georges Haddad

Authors
  1. Georges Haddad
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

B. Cornet V. H. Nguyen J. P. Vial

Rights and permissions

Reprints and permissions

Copyright information

© 1987 The Mathematical Programming Society, Inc.

About this chapter

Cite this chapter

Haddad, G. (1987). The role of tangent and normal cones in the viability theory of differential inclusions. In: Cornet, B., Nguyen, V.H., Vial, J.P. (eds) Nonlinear Analysis and Optimization. Mathematical Programming Studies, vol 30. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0121153

Download citation

  • DOI: https://doi.org/10.1007/BFb0121153

  • Received:

  • Revised:

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00930-3

  • Online ISBN: 978-3-642-00931-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation