Abstract
The generic uniqueness of saddle point for two-person zero-sum differential games, within the class of open-loop, against the perturbation of the right-hand side function of the control system is investigated. By employing set-valued mapping theory, it is proved that the majority of the two-person zero-sum differential games have unique saddle point in the sense of Baire’s category.
1 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.
2 Model and preliminaries
We begin with classical differential games governed by ordinary equations. Let
where
Under some mild conditions, for initial pair
Remark 2.1
It is obvious that
We now introduce the following cost functionals which measures the performance of the control
for some given maps
Problem (DG). For a given initial pair (
Any
Now, we let cost functionals (3) satisfies
where
In this case, Problem(DG) is a two-person zero-sum differential game. For convenience, we call it
and
This yields that
and
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
Definition 2.1
Let initial pair
In this paper,
We make the following assumptions.
[F] The map
[H1] The maps
[H2] For
for all
[I] The following condition holds for any
Remark 2.3
Under the assumptions [F], [I], and [H1]–[H2], Problem(ZDG) admits open-loop saddle point (see [6,7, 8] and references therein).
Next, we state some property on saddle point.
Property 2.1
Let
Proof
Let
Similarly, we can prove that
From the above, (7) holds.
Conversely, let
So,
i.e.,
This completes the proof.□
Property 2.2
Let
Proof
Since
We denote
It follows from (11)) that
Namely, (8) holds. From (8) and (11), we obtain that
for any
3 Generic uniqueness
To investigate the generic uniqueness of open-loop saddle point for Problem(ZDG), we construct the following model. Let
We denote the following set of open-loop saddle points of Problem(ZDG).
Then, the correspondence
Then, one can easily prove that
Next, we recall a series of definitions on set-valued mapping from [21] to study the generic uniqueness of Problem(ZDG).
Let
Lemma 3.1
[22] Let set-valued mapping
Lemma 3.2
[23] Let
Remark 3.1
Let
In what follows, inspired by the literature [18] and [20], we give some basic property about continuous dependence for state trajectory.
Property 3.1
Let
Proof
For any
Since
Owing to
There exists constant
Therefore, choose
Thanks to Gronwall’s inequality, we have
From the arbitrary of
From Property 3.1, the following result is easily obtained.
Corollary 3.1
Let
For any
For any
For any
Corollary 3.2
Let
For any
For any
For any
Now, we present the main results in this paper.
Theorem 3.1
Set-valued mapping
Proof
Since
By
Since
Therefore, for any
which yields
Theorem 3.2
There exists a dense residual subset
Proof
Since
Assume that
Then
Let
where
On the other hand, since
which is a contradiction with
4 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 two-person 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
Acknowledgements
The authors would like to express their sincere thanks to the editor and the anonymous reviewers for their helpful comments and suggestions.
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Funding information: This work was supported by the Natural Science Foundation of China (Grant No. 12061021).
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Author contributions: All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Conflict of interest: Authors state no conflict of interest.
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Data availability statement: Data sharing is not applicable to this paper as no data sets were generated or analyzed during the current study.
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