A survey of viability theory

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 meth...