Generic uniqueness of saddle point for two - person zero - sum di ﬀ erential games

: The generic uniqueness of saddle point for two - person zero - sum di ﬀ erential games, within the class of open - loop, against the perturbation of the right - hand side function of the control system is inves -tigated. By employing set - valued mapping theory, it is proved that the majority of the two - person zero - sum di ﬀ erential games have unique saddle point in the sense of Baire ’ s category.


Introduction
In the 1950s, Isaacs [1] initiated the study of two-person zero-sum differential games. Later in the 1960s and 1970s, Berkovitz [2], Elliott-Kalton [3], Fleming [4], and Friedman [5] also made contributions. Two-person zero-sum differential games were investigated extensively in the literature as they are widely used in many fields, such as biology, finance, and engineering, and also play a key role in the research of general differential games. Ramaswamy and Shaiju [6] proved convergence theorems for the approximate value functions by Yosida type approximations and constructed approximate saddle-point strategies within the sense of feedback in Hilbert Space. Berkovitz [7] defined differential games of fixed duration and showed that games of fixed duration that satisfy Isaacs condition have saddle point. Ghosh and Shaiju [8] proved the existence of saddle point equilibrium for two-player zero-sum differential games in Hilbert space. Ammar et al. [9] derived sufficient and necessary conditions for an open-loop saddle point of rough continuous differential games for two-person zero-sum rough interval continuous differential games. In particular, Sun [10] derived a sufficient condition of the existence of an open-loop saddle point for two-person zero-sum stochastic linear quadratic differential games in 2021. We refer the reader to [11,12] and references therein.
It is worth noting that uniqueness is important in both practice and theory, especially in mathematical problems including two-person zero-sum differential games. However, how many problems have a unique solution? In fact, most mathematical problems cannot guarantee the uniqueness of the solution. So, we have to settle for the second thing: generic uniqueness (see Remark 3.1).
Regarding the generic uniqueness, many results have been investigated. Kenderov [13] studied the solutions of optimization problems and obtained an important result: most optimization problems have a unique solution. Ribarska and Kenderov [14] in their work proved that most two-person zero-sum continuous games have a unique solution in the sense of Baire's category. Tan et al. [15] studied the saddle point for general functions and derived the generic uniqueness of saddle points by the set-valued analysis method. Yu et al. [16] considered the generic uniqueness of equilibrium points for general equilibrium problems.
On the other hand, Yu et al. [17] presented the existence and stability of optimal control problems using set-valued analysis theory in 2014 and showed that most of the optimal control problems are generic stable. After that, Deng and Wei [18,19] proved that generic stability result of optimal control problems governed by semi-linear evolution equation and nonlinear optimal control problems with 1-mean equilibrium controls, respectively. In 2020, the generic stability of Nash equilibria is investigated by Yu and Peng in their work [20] on noncooperative differential games in the sense of Baire's category.
To the best of our knowledge, there is no published result for the generic uniqueness of saddle point for two-person sum-person differential games. The purpose of this paper is to study such problems. We point out that the main idea of the present paper comes from the works of Kenderov [13], Ribarska and Kenderov [14], and Yu et al. [15,20].
The remainder of this paper is organized as follows. The next section is devoted to formulating the game model, collecting some basic preliminary, and stating some properties of a saddle point. In Section 3, we formulate a space of problem and introduce a set-valued mapping. We then state some continuous dependence of state trajectory and cost functional and present some main results in this paper. Finally, some conclusions are given in Section 4.

Model and preliminaries
We begin with classical differential games governed by ordinary equations. Let R p and R q be Euclidean space, U R p ⊂ and V R q ⊂ be bounded closed and convex set. Let T 0 > , for initial state x R n 0 ∈ , consider the following control systems: Under some mild conditions, for initial pair x 0, 0 ( ) and any u v ⋅ ∈ × , control system (1) admits a unique solution.
Remark 2.1. It is obvious that X( ) ⋅ , which is the solution of control system (1), depends on f , u, and v. Thus, let X ⋅ . See the below section for more description with respect to continuous dependence. We now introduce the following cost functionals which measures the performance of the control for some given maps φ The following two-person differential games is posed.
and Player 2 finds is called an open-loop Nash equilibrium control. Now, we let cost functionals (3) satisfies → is the given mapping. Then, one has In this case, Problem(DG) is a two-person zero-sum differential game. For convenience, we call it Remark 2.2. In this paper, our objective is to investigate generic uniqueness of Problem(ZDG) against the perturbation of the right-hand side function of control system. To this end, we assume that cost functional is linear with regard to u( ) ⋅ and v( ) ⋅ , which does not impact our main idea.
In this paper, ‖⋅‖ represents a Euclidean norm.
We make the following assumptions.
Next, we state some property on saddle point.
Similarly, we can prove that From the above, (7) holds.

Generic uniqueness
Then, the correspondence f E f ( ) → yields a set-valued mapping E : Ω 2 U V → × . We shall study the generic uniqueness of E f ( ). The associated metric d R Then, one can easily prove that d Ω, ( ) is a complete metric space. Next, we recall a series of definitions on set-valued mapping from [21] to study the generic uniqueness of Problem(ZDG).
Let U V × be a metric space. A set-valued mapping E : is the graph of Ω. Also recall that a subset Q Ω ⊂ is called a residual set iff it contains countably many intersections of open and dense subsets of Ω. If Ω is a complete space, any residual subset of Ω must be dense in Ω and it is a second category set.

Lemma 3.2. [23]
Let Ω be a complete metric space, U V × be a metric space, and E : Ω 2 U → be an usc mapping with compact. Then there exists a dense residual subset Q of Ω such that E is lower semi-continuous at every point in Q.
⊂ be a dense residual set, if for any β Q ∈ , a certain property P depending on β holds. Then P is called generic property on Ω. Since Q is a second category, we may say that the property P holds for most of the points of Ω in the sense of Baire's category.
In what follows, inspired by the literature [18] and [20], we give some basic property about continuous dependence for state trajectory.
Proof. For any t T 0, , according to control system (1), we have . Thus, f is uniformly continuous on the set Thanks to Gronwall's inequality, we have , and u v u v , , . Without loss of generality, let u u 1 2 ≠ . By separation theorem of convex set, there exists continuous linear functional η in E such that η u η u , let g U R : → be defined by g u η u η u η u η u u U , for any .
Since set-valued mapping E is lower semi-continuous, thus, when ε 0 > is very small, we have G E f ε ( ) ∩ ≠∅ .

Conclusion
By constructing a complete metric space, based on the theory of set-valued mappings, this paper investigates the generic uniqueness of saddle point with respect to the right-hand side functions of the control system for two-person zero-sum differential games within the class of open-loop. That is, most of the twoperson zero-sum differential games have unique saddle point in the sense of Baire's category. However, it is great that our cost functional is linear with respect to control functions u( ) ⋅ and v( ) ⋅ . We will investigate the corresponding stability for a general cost functional in the future.
Acknowledgements: The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
Funding information: This work was supported by the Natural Science Foundation of China (Grant No. 12061021).
Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.