A Viability Theorem for Morphological Inclusions

The aim of this paper is to adapt the viability theorem from differential inclusions (governing the evolution of vectors in a finite-dimensional space) to so-called morphological inclusions (governing the evolution of nonempty compact subsets of the Euclidean space). In this morphological framework, the evolution of compact subsets of $\mathbb{R}^N$ is described by means of flows along bounded Lipschitz vector fields (similarly to the velocity method (a.k.a. speed method) in shape analysis). Now for each compact subset, more than just one vector field is admitted—correspondingly to the set-valued map of a differential inclusion in finite dimensions. We specify sufficient conditions on the given data such that for every initial compact set, at least one of these compact-valued evolutions satisfies fixed state constraints in addition. The proofs follow an approximative track similar to the standard approach for differential inclusions in $\mathbb{R}^N$, but they use tools about weak compactness and weak convergence of Banach-valued functions. Finally an application to shape optimization under state constraints is sketched.