# Multitime hybrid differential games with curvilinear integral functional

## TL;DR

This paper advances the theory of multitime differential games with curvilinear integral functionals, establishing fundamental properties, viscosity solutions, and representation formulas for associated PDEs in a multitime setting.

## Contribution

It provides original results on upper and lower value functions, viscosity solutions, and representation formulas for multitime hybrid differential games constrained by an m-flow.

## Key findings

- Established fundamental properties of multitime upper and lower values.
- Derived viscosity solutions for multitime Hamilton-Jacobi-Isaacs PDEs.
- Developed representation formulas for viscosity solutions of multitime PDEs.

## Abstract

Multitime differential games are related to the modeling and analysis of cooperation or conflict in the context of a multitime dynamical systems. Their theory involves either a curvilinear integral functional or a multiple integral functional and an $m$-flow as constraint. The aim of this paper is to give original results regarding multitime hybrid differential games with curvilinear integral functional constrained by an $m$-flow: fundamental properties of multitime upper and lower values, viscosity solutions of multitime (HJIU) PDEs, representation formula of viscosity solutions for multitime (HJ) PDEs, and max-min representations.