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Viability Theorems for Differential Inclusions

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Viability Theory

Part of the book series: Systems & Control: Foundations & Applications ((MBC))

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Abstract

This is the basic chapter of this book, where the main viability theorems for differential inclusions in finite dimensional vector spaces are gathered and proved. (Invariance Theorems are the topic of Chapter 5.)

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References

  1. BOULIGAND G. (1932) Sur la semi-continuité d’inclusions et quelques sujets connexes,Enseignement Mathématique, 31, 14–22

    Google Scholar 

  2. BOULIGAND G. (1932) INTRODUCTION À LA GÉOMÉTRIE INFINITÉSIMALE DIRECTE, Gauthier-Villars

    Google Scholar 

  3. ZAREMBA S.C. (1936) Sur les équations au paratingent,Bull. Sc. Math., 60, 139–160

    Google Scholar 

  4. MARCHAUD H. (1934) Sur les champs de demi-cônes et les équations différentielles du premier ordre,Bull. Sc. Math., 62, 1–38

    Google Scholar 

  5. CROQUET G. (1947) Convergences,Annales de l’ Univ. de Grenoble, 23, 55–112

    Google Scholar 

  6. MARTIN R.M. (1976) Nonlinear operators and differential equations in Banach spaces, Wiley Interscience, New York

    MATH  Google Scholar 

  7. AUBIN J.-P., FRANKOWSKA H. (1990) SET-VALUED ANALYSIS, Birkhäuser, Systems and Control: Foundations and Applications

    Google Scholar 

  8. AUBIN J.-P., CLARKE (1977) Monotone invariant solutions to differential inclusions, J. London Math. Soc., 16, 357–366

    Google Scholar 

  9. AUBIN J.-P. (1982) Comportement lipschitzien des solutions de problèmes de minimisation convexes,Comptes-Rendus de l’Académie des Sciences, PARIS, 295, 235–238

    Google Scholar 

  10. ROCKAFELLAR R.T. (1979) Clarke’s tangent cones and the boundaries of closed sets in Rn,Nonlinear Anal. Theor. Math. Appl., 3, 145–154

    Google Scholar 

  11. AUBIN J.-P., FRANKOWSKA H. (1987) On the inverse function theorem, J. Math. Pures Appliquées, 66, 71–89

    Google Scholar 

  12. MADERNER N. (to appear) Regulation of control systems under inequality viability constraints

    Google Scholar 

  13. FILIPPOV A.F. (1967) Classical solutions of differential equations with multivalued right hand side,SIAM J. on Control, 5, 609–621

    Google Scholar 

  14. AUBIN J.-P., CELLINA A. (1984) DIFFERENTIAL INCLUSIONS, Springer-Velag, Grundlehren der math. Wiss. # 264

    Chapter  Google Scholar 

  15. FRANKOWSKA H. (1992) CONTROL OF NONLINEAR SYSTEMS AND DIFFERENTIAL INCLUSIONS, Birkhäuser, Systems and Control: Foundations and Applications

    Google Scholar 

  16. FRANKOWSKA H. (1989) Estimations a priori pour les inclusions différentielles opérationelles,Comptes-Rendus de l’Académie des Sciences, PARIS, Série 1, 308, 47–55

    Google Scholar 

  17. FRANKOWSKA H. (1990) A priori estimates for operational differential inclusions, J. Diff. Eqs.

    Google Scholar 

  18. CLARKE F.H. (1975) Generalized gradients and applications,Trans. Am. Math. Soc., 205, 247–262

    Google Scholar 

  19. AUBIN J.-P., CELLINA A. (1984) DIFFERENTIAL INCLUSIONS, Springer-Velag, Grundlehren der math. Wiss. # 264

    Chapter  Google Scholar 

  20. QUINCAMPOIX M. (1990) Frontières de domaines d’invariance et de viabilité pour des inclusions différentielles avec contraintes,Comptes-Rendus de l’Académie des Sciences, Paris, 311, 411–416

    Google Scholar 

  21. QUINCAMPOIX M. (1991) Differential inclusions and target problems,SIAM J. Control,Optimization, IIASA WP-90

    Google Scholar 

  22. QUINCAMPOIX M. (to appear) Enveloppes d’invariance,Cahiers de Mathématiques de la décision

    Google Scholar 

  23. AUBIN J.-P., FRANKOWSKA H., OLECH C. (1986) Controllability of convex processes, SIAM J. of Control and Optimization, 24, 1192–1211

    Google Scholar 

  24. AUBIN J.-P., FRANKOWSKA H., OLECH C. (1986) Contrôlabilité des processus convexes, Comptes-Rendus de l’Académie des Sciences, Paris, 301, 153–156

    Google Scholar 

  25. FRANKOWSKA H. (1990) On controllability and observability of implicit systems,Systems and Control Letters, 14, 219–225

    Google Scholar 

  26. AUBIN J.-P., EKELAND I. (1984) APPLIED NONLINEAR ANALYSIS, Wiley-Interscience

    Google Scholar 

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Aubin, JP. (2009). Viability Theorems for Differential Inclusions. In: Viability Theory. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4910-4_5

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