Detailed Proof of Existence of Value Function in Time-Optimal Games with Lifeline

Abstract

The paper discusses the existence of the value function of a time-optimal differential game with lifeline in a quite general formulation. In such a game, the first player tries to lead the trajectory of the system to a terminal set as soon as possible avoiding the lifeline. The aim of the second player is to lead the trajectory to the lifeline or, if it is not possible, to keep it outside the terminal set, or, if the latter is impossible too, to postpone maximally reaching the terminal set.