# Viscosity solution PDEs in hybrid games with mechanical work payoff

## TL;DR

This paper demonstrates that in a multitime hybrid differential game with mechanical work payoff, the upper and lower value functions are viscosity solutions of the associated Hamilton-Jacobi-Isaacs PDEs, extending the theory of viscosity solutions to complex game settings.

## Contribution

It establishes the viscosity solution property for value functions in multitime hybrid differential games with mechanical work payoff, linking game theory and PDE analysis.

## Key findings

- Value functions are viscosity solutions of Hamilton-Jacobi-Isaacs PDEs.
- Extension of viscosity solution theory to multitime hybrid games.
- Framework for analyzing complex differential games with mechanical work payoff.

## Abstract

In a multitime hybrid differential game with mechanical work payoff, the multitime upper value function and the multitime lower value function are viscosity solutions of original PDEs of type Hamilton-Jacobi-Isaacs.