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
TypeArticle accepté pour publication ou publié
Journal nameSIAM Journal on Control and Optimization
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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 / Keywordsviability; 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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