Viability and Invariance of Systems on Metric Spaces
Résumé
In some applied models (as for instance of flocking or of the crowd control) it is more natural to deal with elements of a metric space (as for instance a family of subsets of a vector space endowed with the Hausdorff metric) rather than with vectors of a normed vector space. 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. As examples of application we study controlled continuity equations on the metric space of probability measures, endowed with the Wasserstein distance, and controlled morphological systems on the space of nonempty compact 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 a proper metric space.
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