Abstract
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 : K → Z 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.
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Keywords
- Differential Inclusion
- Viable Solution
- Contingent Cone
- Nonnegative Continuous Function
- Contingent Derivative
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Bibliographical
AUBIN J.-P. (1982) Comportement lipschitzien des solutions de problèmes de minimisation convexes, Comptes-Rendus de l’Académie des Sciences, PARIS, 295, 235–238
AUBIN J.-P. (1984) Lipchitz behavior of solutions to convex minimization problems, Mathematics of Operations Research, 8, 87–111
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, 1985
AUBIN J.-P. & FRANKOWSKA H. (to appear) Viability kernels of control systems,in NONLINEAR SYNTHESIS, Eds. Byrnes & Kurzhanski, Birkhäuser
MADERNER N. (to appear) Regulation of control systems under inequality viability constraints
CORNET B. & HADDAD G. (1985) Viability theorems for second order differential inclusions, Israel J. Math.
BEBERNES J. W. & KELEY W. (1963) Some boundary value problems for generalized differential equations, SIAM J. Applied Mathematics, 25, 16–23
AUBIN J.-P. & FLIESS M. (in preparation) Ramp controls and polynomial open-loop controls
AUBIN J.-P. & FRANKOWSKA H. (1984) Trajectoires lourdes de systèmes contrôlés, Comptes-Rendus de l’Académie des Sciences, PARIS, 298, 521–524
AUBIN J.-P. & FRANKOWSKA H. (1985) Heavy viable trajectories of controlled systems, Annales de l’Institut HeanriPoincaré, Analyse Non Linéaire, 2, 371–395
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Aubin, JP. (2009). Smooth and Heavy Viable Solutions. In: Viability Theory. Systems & Control: Foundations & Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4910-4_9
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DOI: https://doi.org/10.1007/978-0-8176-4910-4_9
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-4909-8
Online ISBN: 978-0-8176-4910-4
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