On systems of differential equations with constraints in the form of not necessarily convex sets (Q2680509)

Systems with diode nonlinearities (SDNs) are a model of processes confined in a given set of the \(n\)-dimensional space whose behavior inside the bounding set is described by a system of differential equations. Since bounding sets can be nonconvex (for example, when moving under the action of a current in a river or sea basin), it is of interest to generalize the SDN model to a wider class of sets than convex ones. This article is about extending the concept of SDN to some class of not necessarily convex sets. Some properties are established for the extended concept of a normal cone of points in a nonconvex set, and the properties of the projection operator in a neighborhood of the set are also investigated. The relationship between the normal cone and the projection operator is studied. Further, two types of definitions of SDN and definitions of their solutions are given, and the equivalence of two representations of SDNs is proved. A local existence theorem is proved for a solution of the initial value problem for the extended concept of SDN.