Multitime hybrid differential games with multiple integral functional

Abstract

A multiple integral functional is equivalent to a curvilinear integral functional, if the domain is a hyper-parallelepiped, but equivalence is only theoretical. The introduction of this kind of functionals in multitime optimal control problems, particularly in multitime differential games, is due to recent works of Udriste research group. The purpose of this paper is to introduce those ingredients that are necessary to formulate and to prove theorems about multitime differential games based on a multiple integral functional and an $m$-flow as constraint. The most important idea is to use a generating vector field for basic functions. The original results include: fundamental properties of multitime upper and lower values, viscosity solutions of multitime (dHJIU) PDEs, representation formula of viscosity solutions for a multitime (dHJ) PDE, and max-min representations.