Viability and invariance of systems on metric spaces

Abstract

We consider a generalized control system on a metric space and investigate necessary and sufficient conditions for viability and invariance of proper subsets, describing state constraints. Viability means that for every initial condition in the set of constraints we can find trajectories of control system starting at this condition and satisfying state constraints forever. Invariance means that every trajectory of control system starting in the set of constraints never violates them. As examples of application we consider controlled continuity equations on the metric space of Borel probability measures having compact support, endowed with the Wasserstein distance, and controlled morphological systems on the space of nonempty compacts subsets of the Euclidean space endowed with the Hausdorff metric. We also provide sufficient conditions for the existence and uniqueness of contingent solutions to the Hamilton–Jacobi–Bellman equation on proper metric spaces.