Some Problems On The Two-player Zero Sum Differential Game In Hilbert Spaces

Two-player zero sum differential game is mainly concerned with the two-player con-flict of systems driven by differential equations.In recent years,differential game theory has more and more applications in economic,military,social management and so on.Firstly,the information structure of the two-player differential game discussed in this paper is incomplete information,that is,only one of the two players knows the initial state X₀,the other is unknown,but both players know the initial probability measure?₀.This paper mainly proves the existence of the value function of two-player differential game in Hilbert space,and characterizes the value function by the viscous solutions of Hamilton-Jacobi-Isaacs equation in Hilbert space.For this purpose,the definition of value functions is adjusted and some knowledge about viscous solution of Hamilton-Jacobi-Isaacs equation is applied.At the same time,the Wasserstein distance is further improved by using the generalized distance related to probability measure.It is worth noting that this paper weakens the hypotheses of the global Lipschitz contraction of the function f and the continuity of g.Only the function f and the terminal payoff function g satisfy the generalized local contraction.Therefore,in this paper we improve and generalize some corresponding results in the existing literature.Secondly,we consider the stability of Nash equilibria under the condition of corre-sponding differential game.By considering some suitable conditions,a complete metric space(subspace of totally ordered Hilbert space)is constructed,in which each function determines the state equation of two person zero sum differential game.On the basis of introducing the concept of essential equilibria and using the theory of the set-valued anal-ysis,it is proved that the differential game whose Nash equilibrium points are all essential constitutes a dense residual set.Therefore,every differential game can be approximated arbitrarily by a series of essential differential games.Finally,the main work of this paper is summarized,and the future work and possible research directions are prospected.