Infinite dimensional differential games with hybrid controls

TL;DR

This paper investigates a complex infinite-dimensional differential game involving hybrid controls, establishing the existence of a value and characterizing it through viscosity solutions of quasi-variational inequalities.

Contribution

It introduces a novel framework for infinite-dimensional differential games with hybrid controls and proves the existence and characterization of the game's value.

Findings

Existence of the game's value proven.

Value characterized as a unique viscosity solution.

Framework for hybrid controls in infinite-dimensional settings.

Abstract

A two-person zero-sum infinite dimensional differential game of infinite duration with discounted payoff involving hybrid controls is studied. The minimizing player is allowed to take continuous, switching and impulse controls whereas the maximizing player is allowed to take continuous and switching controls. By taking strategies in the sense of Elliott--Kalton, we prove the existence of value and characterize it as the unique viscosity solution of the associated system of quasi-variational inequalities.