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A Survey of Viability Theory

Aubin, Jean-Pierre (1990), A Survey of Viability Theory, SIAM Journal on Control and Optimization, 28, 4, p. 749-788. http://dx.doi.org/10.1137/0328044

Type
Article accepté pour publication ou publié
Date
1990
Journal name
SIAM Journal on Control and Optimization
Volume
28
Number
4
Publisher
SIAM
Pages
749-788
Publication identifier
http://dx.doi.org/10.1137/0328044
Metadata
Show full item record
Author(s)
Aubin, Jean-Pierre
Abstract (EN)
Some theorems of viability theory which are relevant to nonlinear control problems with state constraints and state-dependent control constraints are motivated and surveyed. They all deal with viable solutions to nonlinear control problems, i.e., solutions satisfying at each instant given state constraints of a general and diverse nature. Some classical results on controlled invariance of smooth nonlinear systems are adopted to the nonsmooth case, including inequality constraints bearing on the state and state-dependent constraints on the controls. For instance, existence of a viability kernel of a closed set (corresponding to the largest controlled invariant manifold) is provided under general conditions, even when the zero-dynamics algorithm does not converge. The concepts of slow and heavy viable solutions are introduced, providing concrete ways of regulating viable solutions, by closed-loop feedbacks and closed-loop dynamical feedbacks. Viability theorems also allow the extension of Lyapunov’s second method to nonsmooth observation functions and the construction of “best” Lyapunov functions. As an application, “fuzzy differential inclusion” is presented. Proofs and complements can be found in [Viability Theory, to appear, 1991]. They rely on properties of differential inclusion (see [Differential Inclusions, Springer-Verlag, Berlin, New York, 1984]) and set-valued analysis, (see [Set-Valued Analysis, Birkhäuser, Basel, 1990]).
Subjects / Keywords
viability; invariance; controlled invariance; set-valued maps; regulation map; differential inclusion; fuzzy differential inclusion; Lyapunov stability; asymptotic stability; tracking; contingent cone; contingent derivative of a set-valued map; epicontingent derivative of a function

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