Invariance of sets under mutational inclusions on metric spaces - Frankowska, Hélène; Lorenz, Thomas (Nonlinear differential equations and applications, 2023)
This paper is devoted to the invariance of time dependent sets under dynamical systems defined on a metric space. In the absence of vector structure, evolutions are described by the so-called mutational inclusions, that extend differential inclusions of the classical Euclidean framework to the one of general metric spaces. The main difficulty we have to overcome is the absence of local compactness of the constraints implying that the distance between a point and a closed set, in general, is not realized. The fairy technical proof we propose has a potential to be applicable also to problems of invariance of closed sets under various evolution laws. The reason to seek such generality lies in the desire to have universal results that can be applied to various settings, as for instance to classical differential inclusions, to continuity inclusions in the Wasserstein spaces, the controlled transport equation in the space of Radon measures or morphological systems in the space of closed subsets of a Banach space. The obtained results are illustrated by the example of dynamics described by a non-homogeneous controlled continuity equation on the space of Radon measures.