Generic uniqueness of saddle point for two-person zero-sum differential games

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 ( ⋅ ) ≡ X u , v f ( ⋅ ) . 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 u ( ⋅ ) and v ( ⋅ ) .

for some given maps φ i : [ 0 , T ] × R n × U × V → R and ψ i : R n → R ( i = 1 , 2 ) . The following two-person differential games is posed.

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.

Definition 2.1

Let initial pair ( 0 , x 0 ) be fixed. A control pair ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] is called an open-loop saddle point of Problem(ZDG), if for any ( u ( ⋅ ) , v ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] , it satisfies

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).

Property 2.1

Let ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] . Then ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) is a saddle point of Problem(ZDG) if and only if (for short, iff)

Proof

Let ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) be a saddle point, then for any u ( ⋅ ) ∈ U [ 0 , T ] , we have J ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) ≤ J ( u ( ⋅ ) , v ¯ ( ⋅ ) ) . This implies that J ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) ≤ sup v ( ⋅ ) ∈ V [ 0 , T ] J ( u ( ⋅ ) , v ( ⋅ ) ) , which results in

Similarly, we can prove that

From the above, (7) holds.

Conversely, let ω = J ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) , that is ω = inf u ( ⋅ ) ∈ U [ 0 , T ] sup v ( ⋅ ) ∈ V [ 0 , T ] J ( u ( ⋅ ) , v ( ⋅ ) ) = sup v ( ⋅ ) ∈ V [ 0 , T ] inf u ( ⋅ ) ∈ U [ 0 , T ] J ( u ( ⋅ ) , v ( ⋅ ) ) . Then for any u ( ⋅ ) ∈ U [ 0 , T ] and v ( ⋅ ) ∈ V [ 0 , T ] , we have

So,

i.e.,

This completes the proof.□

Property 2.2

Let ( u ¯ 1 ( ⋅ ) , v ¯ 1 ( ⋅ ) ) , ( u ¯ 2 ( ⋅ ) , v ¯ 2 ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] be saddle point of Problem(ZDG). Then ( u ¯ 1 ( ⋅ ) , v ¯ 2 ( ⋅ ) ) , ( u ¯ 2 ( ⋅ ) , v ¯ 1 ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] are also saddle point and

Proof

Since ( u ¯ 1 ( ⋅ ) , v ¯ 1 ( ⋅ ) ) , ( u ¯ 2 ( ⋅ ) , v ¯ 2 ( ⋅ ) ) are saddle points, then for any ( u ( ⋅ ) , v ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] , we have

We denote u ( ⋅ ) = u ¯ 2 ( ⋅ ) , v ( ⋅ ) = v ¯ 2 ( ⋅ ) , and u ( ⋅ ) = u ¯ 1 ( ⋅ ) , v ( ⋅ ) = v ¯ 1 ( ⋅ ) in (9) and (10), respectively.

It follows from (11)) that

Namely, (8) holds. From (8) and (11), we obtain that

for any ( u ( ⋅ ) , v ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] . This completes the proof.□

Lemma 3.1

[22] Let set-valued mapping E : Ω → 2 U × V be closed and U × V be compact, then E is upper semi-continuous at each f ∈ Ω .

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 .

Remark 3.1

Let Q ⊂ Ω 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.

Property 3.1

Let { f k } ⊂ Ω with f k → f ∈ Ω . For any ( u k ( ⋅ ) , v k ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] with ( u k ( ⋅ ) , v k ( ⋅ ) ) → ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] , one has X u k , v k f k ( ⋅ ) → X u ¯ , v ¯ f ( ⋅ ) as k → ∞ .

Proof

For any t ∈ [ 0 , T ] , according to control system (1), we have

Since f k → f , for any ε > 0 , there exists N 1 > 0 such that for any k > N 1 , d ( f k , f ) < ε 3 T . X ( t ) is continuous at [ 0 , T ] , then there exists constant a 1 > 0 such that max t ∈ [ 0 , T ] ∥ X ( t ) ∥ ≤ a 1 . U ⊂ R p and V ⊂ R q are bounded closed and convex set. That is, U and V are also compact. Because u ( ⋅ ) and v ( ⋅ ) are continuous in [ 0 , T ] , there exist constants a 2 > 0 and a 3 > 0 such that max t ∈ [ 0 , T ] ∥ u ( t ) ∥ ≤ a 2 and max t ∈ [ 0 , T ] ∥ v ( t ) ∥ ≤ a 3 . Thus, f is uniformly continuous on the set

Owing to ( u k ( ⋅ ) , v k ( ⋅ ) ) → ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) , there exists constant N 2 > 0 such that for any t ∈ [ 0 , T ] , when k ≥ N 2 , one has

There exists constant N 3 > 0 such that for any t ∈ [ 0 , T ] , when k ≥ N 3 , one has

Therefore, choose N = max { N 1 , N 2 , N 3 } such that for any t ∈ [ 0 , T ] , when k ≥ N , one has

Thanks to Gronwall’s inequality, we have

From the arbitrary of ε > 0 , it yields X u k , v k f k ( ⋅ ) → X u ¯ , v ¯ f ( ⋅ ) .□

Corollary 3.1

Let { f k } ⊂ Ω with f k → f ∈ Ω .

1. For any u k ( ⋅ ) ∈ U [ 0 , T ] with u k ( ⋅ ) → u ¯ ( ⋅ ) ∈ U [ 0 , T ] . Then for any v ( ⋅ ) ∈ V [ 0 , T ] , X u k , v f k ( ⋅ ) → X u ¯ , v f ( ⋅ ) as k → ∞ .

2. For any v k ( ⋅ ) ∈ V [ 0 , T ] with v k ( ⋅ ) → v ¯ ( ⋅ ) ∈ V [ 0 , T ] . Then for any u ( ⋅ ) ∈ U [ 0 , T ] , X u , v k f k ( ⋅ ) → X u , v ¯ f ( ⋅ ) as k → ∞ .

3. For any ( u ( ⋅ ) , v ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] . Then X u , v f k ( ⋅ ) → X u ¯ , v ¯ f ( ⋅ ) as k → ∞ .

Corollary 3.2

Let f ∈ Ω .

1. For any ( u k ( ⋅ ) , v k ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] with ( u k ( ⋅ ) , v k ( ⋅ ) ) → ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) ∈ U [ 0 , T ] × V [ 0 , T ] . Then J f ( u k ( ⋅ ) , v k ( ⋅ ) ) → J f ( u ¯ ( ⋅ ) , v ¯ ( ⋅ ) ) as k → ∞ .

2. For any u k ( ⋅ ) ∈ U [ 0 , T ] with u k ( ⋅ ) → u ¯ ( ⋅ ) ∈ U [ 0 , T ] . Then for any v ( ⋅ ) ∈ V [ 0 , T ] , J f ( u k ( ⋅ ) , v ( ⋅ ) ) → J f ( u ¯ ( ⋅ ) , v ( ⋅ ) ) as k → ∞ .

3. For any v k ( ⋅ ) ∈ V [ 0 , T ] with v k ( ⋅ ) → v ¯ ( ⋅ ) ∈ V [ 0 , T ] . Then for any u ( ⋅ ) ∈ U [ 0 , T ] , J f ( u ( ⋅ ) , v k ( ⋅ ) ) → J f ( u ( ⋅ ) , v ¯ ( ⋅ ) ) as k → ∞ .