# Multitime hybrid differential games with multiple integral functional

## TL;DR

This paper develops a theoretical framework for multitime differential games involving multiple integral functionals, introducing key properties, viscosity solutions, and representation formulas within this complex multitime setting.

## Contribution

It introduces the necessary ingredients and proves fundamental theorems for multitime differential games based on multiple integral functionals and $m$-flows, expanding the theoretical foundation.

## Key findings

- Properties of multitime upper and lower values established
- Viscosity solutions for multitime (dHJIU) PDEs derived
- Representation formulas for multitime (dHJ) PDEs provided

## 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.