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Set-Valued Maps

  • Jean-Pierre Aubin
Part of the Systems & Control: Foundations & Applications book series (MBC)

Abstract

We shall gather in this chapter some of the results dealing with set-valued maps that we shall need. Only the properties of upper semicontinuous set-valued maps and, among them, the Convergence Theorem 2.4.4, and some notions on the set-valued analogues of continuous linear operators, the closed convex processes are required in the short term. Hence, further results, in particular those dealing with lower semicontinuous criteria and selections of lower semicontinuous maps, are postponed to Chapter 6.

Keywords

Banach Space Lower Semicontinuous Continuous Linear Operator Separation Theorem Marginal Function 
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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References

  1. [14]
    AUBIN J.-P., FRANKOWSKA H., OLECH C. (1986) Controllability of convex processes, SIAM J. of Control and Optimization, 24, 1192–1211Google Scholar
  2. [15]
    AUBIN J.-P., FRANKOWSKA H., OLECH C. (1986) Contrôlabilité des processus convexes, Comptes-Rendus de l’Académie des Sciences, Paris, 301, 153–156Google Scholar
  3. [49]
    AUBIN J.-P. (1987) Smooth and heavy solutions to control problems,in NONLINEAR AND CONVEX ANALYSIS, Eds. BL. Lin,Simons S., Proceedings in honor of Ky Fan, Lecture Notes in pure and applied mathematics, June 24–26, 1985Google Scholar
  4. [298]
    KRENER A.,ISIDORI A. (1980) Nonlinear zero distributions,19th IEEE Conf. Decision and ControlGoogle Scholar
  5. [132]
    BYRNES C.I., ISIDORI A. (1984) A frequency domain philosophy for nonlinear systems, with applications to stabilization and adaptive control, 23rd IEEE Conf. Dec. Control, 15691573Google Scholar
  6. [133]
    BYRNES C.I., ISIDORI A. (1988) Heuristics in nonlinear control and modelling, In ADAPTIVE CONTROL, Byrnes, Kurzhanski Eds., Springer-VerlagGoogle Scholar
  7. [135]
    BYRNES C.I., ISIDORI A. (1990) Feedback design from the zero dynamics point of view, in COMPUTATION AND CONTROL, Bowers K., Lund J. Eds., Birkhäuser, 23–52Google Scholar
  8. [136]
    BYRNES C.I., ISIDORI A. (1990) Output regulation of nonlinear systems, IEEE Trans. Autom. Control, 35, 131–140Google Scholar
  9. [137]
    BYRNES C.I., ISIDORI A. (1990) Régulation asymptotique des systèmes non linéaires, Comptes-Rendus de l’Académie des Sciences, Paris, 309, 527–530Google Scholar
  10. [138]
    BYRNES C.I.,ISIDORI A. (to appear) Asymptotic stabilization of minimum phase nonlinear systems Google Scholar
  11. [139]
    BYRNES C.I., ISIDORI A. (to appear) New methods for shaping the response of a nonlinear system,in NONLINEAR SYNTHESIS, Eds. Byrnes, Kurzhanski, BirkhäuserGoogle Scholar
  12. [380]
    MONACO S., NORMAND-CYROT D. (1988) Zero dynamics of sampled linear systems,Systems and Control LettersGoogle Scholar
  13. [284]
    ISIDORI A.,MOOG C.H. (1988) On the nonlinear equivalent of the notion of transmission zeros,in MODELLING AND ADAPTIVE CONTROL, Byrnes, Lurzhanski Eds. LNCIS 105, SPRINGER, 146–158Google Scholar
  14. [285]
    ISIDORI A. (1989) NONLINEAR CONTROL SYSTEMS, Springer-VerlagGoogle Scholar
  15. [11]
    AUBIN J.-P., BYRNES C., ISIDORI A. (1990) Viability kernels, controlled invariance and zero dynamics for nonlinear systems, Proceedings of the 9th International Conference on Analysis and Optimization of Systems, Nice, June 1990, Lecture Notes in Control and Information Sciences, Springer-VerlagGoogle Scholar
  16. [140]
    BYRNES C.I., ISIDORI A. (to appear) NONLINEAR FEEDBACK DESIGN, BirkhäuserGoogle Scholar
  17. [279]
    HOFBAUER J., SIGMUND K. (1988) THE THEORY OF EVOLUTION AND DYNAMICAL SYSTEMS, Cambridge University Press, London Math. Soc. # 7Google Scholar
  18. [37]
    AUBIN J.-P., SIGMUND K. (1988) Permanence and viability,J. of Computational and Applied Mathematics, 22, 203209Google Scholar
  19. [303]
    KRIVAN V. (to appear) Construction of population growth equations in the presence of viability constraints,J. Math. BiologyGoogle Scholar
  20. [430]
    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, 411–416Google Scholar
  21. [432]
    QUINCAMPOIX M. (1991) Differential inclusions and target problems,SIAM J. Control,Optimization, IIASA WP-90Google Scholar
  22. [463]
    SAINT-PIERRE P. (to appear) Viability property of the boundary of the viability kernel Google Scholar
  23. [284]
    ISIDORI A.,MOOG C.H. (1988) On the nonlinear equivalent of the notion of transmission zeros,in MODELLING AND ADAPTIVE CONTROL, Byrnes, Lurzhanski Eds. LNCIS 105, SPRINGER, 146–158Google Scholar
  24. [285]
    ISIDORI A. (1989) NONLINEAR CONTROL SYSTEMS, Springer-VerlagGoogle Scholar
  25. [281]
    HU X.-M. (to appear) PhD. Thesis, Arizona State UniversityGoogle Scholar
  26. [478]
    SILVERMAN L. M. (1969) Inversion of multivariable linear systems, IEEE Trans. Automatic Control, 14, 270–276Google Scholar
  27. [77]
    BASILE G., MARRO G. (1969) Controlled and conditional invariat subspaces in linear system theory, J.Optim. Theory Appl., 3, 396–315Google Scholar
  28. [382]
    MOORE B.C., LAUB A.J. (1978) Computation of supremal (A, B)-invariant and controllability subspaces,IEEE Trans. Autom. Control, AC-23, 783–792Google Scholar
  29. [239]
    FRANKOWSKA H., QUINCAMPOIX M. (1991) Viability kernels of differential inclusions with constraints, Mathematics of Systems, Estimation and ControlGoogle Scholar
  30. [240]
    FRANKOWSKA H., QUINCAMPOIX M. (1991) Un algorithme déterminant les noyaux de viabilité pour des inclusions différentielles avec contraintes, Comptes-Rendus de l’Académie des Sciences, PARIS, Série 1, 312, 31–36Google Scholar
  31. [177]
    COLOMBO G., KRIVAN V. (to appear) A viability algorithm,Preprint SISSAGoogle Scholar
  32. [35]
    AUBIN J.-P., SAINT-PIERRE P. (to appear) Approximation aux différences finies des noyaux de viabilité,Cahiers de Mathématiques de la DécisionGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2009

Authors and Affiliations

  • Jean-Pierre Aubin
    • 1
  1. 1.EDOMADE (Ecole Doctorale de Mathématique de la Décision)Université de Paris-DauphineParis cedex 16France

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