Smooth and Heavy Viable Solutions

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


Let us still consider the problem of regulating a control system \( \ge 0,x'(t) = f(x(t),u(t)) \) where \( u\left( t \right) \in U\left( {x\left( t \right)} \right) \) where U : KZ associates with each state x the set U(x) of feasible controls (in general state-dependent) and f : Graph(U) ↦ X describes the dynamics of the system.


Differential Inclusion Viable Solution Contingent Cone Nonnegative Continuous Function Contingent Derivative 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [46]
    AUBIN J.-P. (1982) Comportement lipschitzien des solutions de problèmes de minimisation convexes, Comptes-Rendus de l’Académie des Sciences, PARIS, 295, 235–238zbMATHMathSciNetGoogle Scholar
  2. [47]
    AUBIN J.-P. (1984) Lipchitz behavior of solutions to convex minimization problems, Mathematics of Operations Research, 8, 87–111CrossRefMathSciNetGoogle 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. [33]
    AUBIN J.-P. & FRANKOWSKA H. (to appear) Viability kernels of control systems,in NONLINEAR SYNTHESIS, Eds. Byrnes & Kurzhanski, BirkhäuserGoogle Scholar
  5. [368]
    MADERNER N. (to appear) Regulation of control systems under inequality viability constraints Google Scholar
  6. [186]
    CORNET B. & HADDAD G. (1985) Viability theorems for second order differential inclusions, Israel J. Math.Google Scholar
  7. [82]
    BEBERNES J. W. & KELEY W. (1963) Some boundary value problems for generalized differential equations, SIAM J. Applied Mathematics, 25, 16–23Google Scholar
  8. [23]
    AUBIN J.-P. & FLIESS M. (in preparation) Ramp controls and polynomial open-loop controls Google Scholar
  9. [24]
    AUBIN J.-P. & FRANKOWSKA H. (1984) Trajectoires lourdes de systèmes contrôlés, Comptes-Rendus de l’Académie des Sciences, PARIS, 298, 521–524zbMATHMathSciNetGoogle Scholar
  10. [25]
    AUBIN J.-P. & FRANKOWSKA H. (1985) Heavy viable trajectories of controlled systems, Annales de l’Institut HeanriPoincaré, Analyse Non Linéaire, 2, 371–395zbMATHMathSciNetGoogle 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

Personalised recommendations