Dynamic Programming for Nash Equilibrium in Zero-Sum Impulse Control Games

This paper aims to solve the two-party game decision-making problem in a complex dynamic environment, and proposes a deterministic zero-sum impulse differential control model. The two players face the problem of decision-making discontinuity in the confrontation process, and the traditional differential game model cannot effectively portray this behaviour. For this reason, this paper combines the theory of impulse control to model this decision-making process as a zero-sum game model, and the strategy not only includes continuous control, but also introduces impulse control, thus solving the decision discontinuity problem. In addition, this paper combines the Dynamic Programming (DP) with the Hamilton-Jacobi-Belman-Isaacs (HJBI) equations and Quasi-Variational Inequalities (QVIs) to perform a detailed mathematical analysis, proving the uniqueness of the up and down value functions and the existence of Nash equilibrium solutions. The effectiveness of the hybrid game strategy proposed in this paper is verified by numerical simulation.