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A survey of viability theory. (English) Zbl 0714.49021

Summary: 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 the author’s work, “Viability theory” (1991, to appear). They rely on properties of differential inclusion [see the author and A. Cellina, “Differential inclusions. Set-valued maps and viability theory” (1984; Zbl 0538.34007)] and set-valued analysis, [see the author, “Set-valued analysis” (Basel 1990)].

MSC:
49J52Nonsmooth analysis (other weak concepts of optimality)
93C10Nonlinear control systems
26A24Differentiation of functions of one real variable
49-02Research monographs (calculus of variations)
93-02Research monographs (systems and control)
26A27Nondifferentiability of functions of one real variable; discontinuous derivatives
26A51Convexity, generalizations (one real variable)
26E25Set-valued real functions
28B20Set-valued set functions and measures; integration of set-valued functions; measurable selections
28D05Measure-preserving transformations
34A60Differential inclusions
34DxxStability theory of ODE
39AxxDifference equations
54C60Set-valued maps (general topology)
54C65Continuous selections
58C06Set-valued and function-space valued mappings on manifolds
58C07Continuity properties of mappings on manifolds
58C30Fixed point theorems on manifolds
93C15Control systems governed by ODE
93C30Control systems governed by other functional relations