A viability theorem for set-valued states in a Hilbert space

Applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact (possibly non-convex) sets rather than vectors. This leads to evolutions in a basic set which can be supplied with a metric (like the well-established Pompeiu–Hausdorff distance), but it does not have an obvious linear structure. This article extends differential inclusions with state constraints to compact-valued states in a separable Hilbert space H. The focus is on sufficient conditions such that a given constraint set (of compact subsets) is viable a.k.a. weakly invariant. Our main result extends the tangential criterion in the well-known viability theorem (usually for differential inclusions in a vector space) to the metric space of non-empty compact subsets of H.