Zero-sum discounted semi-Markov games with stopping and control

This paper investigates a two-person zero-sum semi-Markov game with stopping and control under the expected discounted reward criterion, where players can both use controls to influence the evolution of the game and choose to stop the game. Under some mild conditions, we prove the Shapley equation has a solution. Using this solution, we establish the existence of the value, and construct a Nash equilibrium consisting of stopping times and stationary controls. Additionally, we develop an iterative algorithm to compute $ \varepsilon $-Nash equilibria. Finally, an example of an energy management system is given to illustrate the applications of our results.