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In this paper, firstly a new class of time-delay differential inequality is proved. Then as an application, the nonlinearly perturbed differential systems with multiple delay are considered and it is obtained that the trivial solution of the nonlinear systems with multiple delay has uniform stability and uniform exponential Lipschitz asymptotic stability with respect to partial variables. It is obvious that the above system is a generalization of the traditional differential systems. The aim of this paper is to investigate the double stability of time-delay differential equations, including Uniform stability and Uniform Lipschitz stability. The author uses the method of differential inequalities with time-delay and integral inequalities to establish double stability criteria. As a result, the partial stability of differential equations is widely used both in theory and in practice such as dynamic systems and control systems.

In 1892, Lyapunov, a Russian mathematician, mechanician and physicist, proposed the notion of the stability of motion. He gave the general research methods in his doctoral dissertation “The general problem of the stability of motion” [

In 1986, Dannan and Elaydi ( [

d x d t = f ( t , x ) , (1)

where f ∈ C [ J × R n , R n ] , J = [ 0 , ∞ ) , f ( t , 0 ) = 0 , and x ( t , t 0 , x 0 ) ≡ x ( t ) is the solution of (1) with x ( t 0 , t 0 , x 0 ) = x 0 , where t 0 ≥ 0 .

This notion of ULS lies somewhere between uniform stability (US) on one side and the notions of asymptotic stability in variation (ASV) and uniform stability in variation (USV) on the other side. An important feature of ULS is that the linearized system inherits the property of ULS from the original nonlinear system.

YU-LI Fu ( [

d x d t = f ( t , x t ) , (2)

where x ∈ R n , f : R × C ( [ − r , 0 ] , R n ) ↦ R n , f ( t , 0 ) = 0 , f is continuous, x t = x ( t + θ ) , θ ∈ [ − r , 0 ] , r > 0 .

Sung Kyu Choi, Ki Shik Koo and Keonhee Lee ( [

d x d t = A ( t ) x + g ( t , x ) , (3)

d x d t = f ( t , x ) + g ( t , x ) , (4)

where A ( t ) is a continuous n × n matrix defined on R + , g ( t , x ) ∈ C ( R + × R n , R n ) with g ( t , 0 ) = 0 .

Vorotnikov, V. I. ( [

{ d y d t = A ( t ) y + B ( t ) z + Y ( t , y , z ) d z d t = C ( t ) y + D ( t ) z + Z ( t , y , z ) , (5)

and studied the double stability as ‖ y ‖ + ‖ z ‖ → 0 and ‖ Y ( t , y , z ) ‖ + ‖ Z ( t , y , z ) ‖ ‖ y ‖ + ‖ z ‖ → 0 .

In this paper, the author considers a new class of the nonlinearly perturbed differential systems with time-delay

d x d t = A ( t ) x + f ( t , x ( t ) , x ( t − τ ) , ∫ 0 t h ( s , x ( s ) , x ( s − τ ) ) d s ) , (6)

where x ∈ R n , y = c o l ( x 1 , x 2 , ⋯ , x m ) , z = c o l ( x m + 1 , x m + 2 , ⋯ , x n ) , x = c o l ( y , z ) , f : R × R n × C ( [ − r , 0 ] , R n ) × R n ↦ R n , f ( t , 0 , 0 , 0 ) ≡ 0 , h : R × R n × C ( [ − r , 0 ] , R n ) ↦ R n , τ is a non-negative constant.

It is obvious that the above system is a generalization of the systems in [

1) Definitions and lemmas

Consider the following system:

d x d t = f ( t , x , x ( t − τ ) ) , (7)

where x ∈ R n , y = c o l ( x 1 , x 2 , ⋯ , x m ) , z = c o l ( x m + 1 , x m + 2 , ⋯ , x n ) , x = c o l ( y , z ) , f ( t , 0 , 0 ) ≡ 0 , τ is a non-negative constant. Let ϕ ( t ) be a continuous function, for ∀ t ∈ E t 0 = [ t 0 − τ , t 0 ] .

Definition 1: The trivial solution of system (7) has uniform stability and exponential asymptotic stability with respect to y if, for ∀ ε > 0 , ∀ t 0 ∈ I , ∃ δ ( ε ) > 0 and λ > 0 , when ‖ ϕ ‖ < δ (for ∀ t ∈ E t 0 ), such that ‖ x ( t ; t 0 , ϕ ) ‖ < ε , ‖ y ( t ; t 0 , ϕ ) ‖ < ε exp ( − λ ( t − t 0 ) ) , for all t ≥ t 0 .

Definition 2: The trivial solution of system (7) has Lipschitz stability with respect to y if, there exist constants M ( t 0 ) > 0 and δ ( t 0 ) > 0 , when ‖ ϕ ‖ < δ (for ∀ t ∈ E t 0 ), such that ‖ y ( t ; t 0 , ϕ ) ‖ ≤ M ( t 0 ) ‖ ϕ ‖ , for all t ≥ t 0 ≥ 0 .

Definition 3: The trivial solution of system (7) has equi-exponential Lipschitz asymptotic stability with respect to y if, there exist constants λ > 0 , K ( t 0 ) > 0 and δ ( t 0 ) > 0 , when ‖ ϕ ‖ < δ (for ∀ t ∈ E t 0 ), such that ‖ y ( t ; t 0 , ϕ ) ‖ ≤ K ( t 0 ) ‖ ϕ ‖ exp ( − λ ( t − t 0 ) ) , for all t ≥ t 0 ≥ 0 .

Definition 4: The trivial solution of system (7) has uniform exponential Lipschitz asymptotic stability with respect to y if, K and δ in Definition3 are unrelated to t 0 .

Lemma 1. [

{ d y d x = B ( t ) y + C ( t ) z d z d x = D ( t ) y + E ( t ) z , (8)

if the trivial solution of system (8) has uniform stability, and has exponential asymptotic stability with respect to y, then there exists a Lyapunov-function V ( t , x ) satisfied the following conditions:

‖ y ‖ ≤ V ( t , x ) ≤ M ‖ x ‖ , V ˙ | ( 8 ) ≤ − α V ( t , x ) ,

where M > 0 .

Consider the following inequality:

x ˙ i ( t ) ≤ f i ( t ) [ − r i x i ( t ) + h i ( 1 ) ( x t ) x t α i + ∫ − ∞ t h i ( 2 ) ( t − s , x ( s ) ) x β i ( t ) e − ε ( t − s ) d s ] , (9)

where f i ( t ) ∈ C [ R , R + ] and f i ( t ) ≥ β = c o n s t > 0 , r i = c o n s t > 0 , ( i = 1 , 2 , ⋯ , m ) , h i ( 1 ) ( ⋅ ) , h i ( 2 ) ( t − θ , ⋅ ) are nonnegative and not monotone decreasing for “ ⋅ ”, α i , β i ≥ 1 , x ( θ ) ≜ max 1 ≤ j ≤ n ( x j ( θ ) ) , x t ≜ max 1 ≤ j ≤ n ( sup t − τ ≤ θ ≤ t x j ( θ ) ) , τ = c o n s t > 0 , α ≜ max ( α i , β i ) .

Lemma 2. [

h i ( 1 ) ( K ) + ∫ 0 + ∞ h i ( 2 ) ( s , K ) d s < r i , α K 1 − 1 α < 1 ,

when M α ≜ max 1 ≤ j ≤ n ( sup t 0 − τ ≤ θ ≤ t 0 x j ( θ ) ) < K , we have following result:

x i ( t ) ≤ M exp ( − λ ( t − t 0 ) ) ,

holds true, where t ≥ t 0 , and λ > 0 .

2) Differential Inequalities with Time-Delay

Consider the following inequality

x ˙ i ( t ) ≤ f i ( t ) [ − r i x i ( t ) + h i ( 1 ) ( x t ) x t α i + ∫ − ∞ t h i ( 2 ) ( t − s , x ( s ) ) x β i ( t ) e − ε ( t − s ) d s ] , (10)

where f i ( t ) ∈ C [ R , R + ] and f i ( t ) ≥ γ = c o n s t > 0 , r i = c o n s t > 0 , h i ( 1 ) ( ⋅ ) , h i ( 2 ) ( t − θ , ⋅ ) ( i = 1 , 2 , ⋯ , n ) are nonnegative and not monotone ecreasing for “ ⋅ ”, α i , β i ≥ 1 , x ( θ ) = max 1 ≤ i ≤ n ( x i ( θ ) ) , x t = max 1 ≤ i ≤ n ( sup t − τ < θ < t x i ( θ ) ) , τ = c o n s t > 0 , α = max ( α i , β i ) .

Lemma 3. Assume x i ( t ) be nonnegative continuous on R + , (10) is satisfied for all t ≥ t 0 , there exists a constant K satisfied the following inequality:

h i ( 1 ) ( K ) + ∫ 0 + ∞ h i ( 2 ) ( s , K ) d s < r i , (11)

and

α K 1 − 1 α < 1 ,

then if

x i ( t ) ≤ M exp ( − λ ( t − t 0 ) ) ( i = 1 , 2 , ⋯ , n )

holds true, where t ≥ t 0 and λ > 0 .

Proof

According to (10), for ∀ ε > 0 , ∃ λ (let λ < ε α ) we can get

− r i + α λ γ + e α λ τ h i ( 1 ) ( K ) + ∫ 0 + ∞ h i ( 2 ) ( s , K ) d s < 0.

Now define

P i ( t ) ≜ { x i α ( t ) e α λ ( t − t 0 ) , t ≥ t 0 ; ( 12 ) x i α ( t ) , t ∈ [ t 0 − τ , t 0 ] , ( i = 1 , 2 , ⋯ , n )

thus, we can have

x i ( t ) = P i 1 α ( t ) e − λ ( t − t 0 ) ,

furthermore

x ˙ i ( t ) = ( 1 α P i 1 α − 1 ( t ) P ˙ i ( t ) − λ P i 1 α ( t ) ) e − λ ( t − t 0 ) . (13)

Let

P t = max 1 ≤ i ≤ n ( sup t − τ < θ < t P i ( θ ) ) , P ( θ ) = max 1 ≤ i ≤ n ( P i ( θ ) ) ,

obviously

P t ≥ x t α , P ( θ ) ≥ x α ( θ ) ,

hence

x ˙ i ( t ) = ( 1 α P i 1 α − 1 ( t ) P ˙ i ( t ) − λ P i 1 α ( t ) ) e − λ ( t − t 0 ) ≤ f i ( t ) [ − r i x i ( t ) + h i ( 1 ) ( x t ) x t α i + ∫ − ∞ t h i ( 2 ) ( t − s , x ( s ) ) x β i ( t ) e − ε ( t − s ) d s ] .

Notice that

x t α = max 1 ≤ i ≤ n ( sup t − τ < θ < t x i α ( θ ) ) = max 1 ≤ i ≤ n ( sup t − τ < θ < t x i α ( θ ) ) = max 1 ≤ i ≤ n ( sup t − τ < θ < t P i ( θ ) e − α λ ( θ − t 0 ) ) = max 1 ≤ i ≤ n ( sup t − τ < θ < t P i ( θ ) ) e − α λ ( t − τ − t 0 ) = P t e − α λ ( t − t 0 ) e α λ τ ,

and

x α ( θ ) = max 1 ≤ i ≤ n ( x i α ( θ ) ) = max 1 ≤ i ≤ n ( P i ( θ ) e − α λ ( θ − t 0 ) ) = max 1 ≤ i ≤ n ( P i ( θ ) e − α λ ( θ − t 0 ) ) = max 1 ≤ i ≤ n ( P i ( θ ) ) e − α λ ( θ − t 0 ) = P ( θ ) e − α λ ( θ − t 0 ) .

Applying (12) into (10), we have

P ˙ i ( t ) ≤ f i ( t ) [ − ( r i − α λ γ ) P i ( t ) + ( h i ( 1 ) ( x t ) P t e α λ τ + ∫ 0 + ∞ h i ( 2 ) ( t − s , x ( s ) ) P ( s ) d s ) α P i 1 − 1 α ( t ) ] . (14)

For any scaler l ∈ ( 1 , K M α ) , we can get

P i ( t ) ≤ l M α ≜ N .

If not, then P i ( t ) < N , thus there exists a certain i in ( − ∞ , t 0 ] and t 1 > t 0 , we have

P i ( t 1 ) = N , P j ( t ) { < N , j = i , t ∈ ( − ∞ , t 1 ) ; ≤ N , j ≠ i , t ∈ ( − ∞ , t 1 ] ,

thus we can get P ˙ i ( t 1 ) ≥ 0 . Using it in (14), we get

P ˙ i ( t 1 ) ≤ f i ( t 1 ) [ − ( r i − α λ β ) P i ( t 1 ) + ( h i ( 1 ) ( x t 1 ) P t 1 e α λ τ + ∫ − ∞ t 1 h i ( 2 ) ( t 1 − s , x ( s ) ) P ( s ) d s ) α P i 1 − 1 α ( t 1 ) ] ≤ f i ( t 1 ) [ − ( r i − α λ β ) K + ( h i ( 1 ) ( K ) K e α λ τ + ∫ 0 + ∞ h i ( 2 ) ( s , K ) K d s ) α K 1 − 1 α ] ≤ f i ( t 1 ) [ − ( r i − α λ β ) + ( h i ( 1 ) ( K ) e α λ τ + ∫ 0 + ∞ h i ( 2 ) ( s , K ) d s ) ] K < 0.

It is a contradictory, thus P i ( t ) ≤ l M α , let l → 1 , we can get

P i ( t ) ≤ M α .

Notice (12), the following is obtained

x i ( t ) ≤ M exp ( − λ ( t − t 0 ) ) , for all t ≥ t 0 .

Remark It is obvious that when α i = 1 , β i = 1 lemma 2 can be deduced by lemma 3.

Consider the following system which is equivalent with system (1)

{ d y d t = B ( t ) y + C ( t ) z + Y ( t , y ( t ) , z ( t ) , ∫ 0 t h 1 ( s , y ( s ) , z ( s ) , y ( s − τ ) , z ( s − τ ) ) d s ) d x d t = D ( t ) y + E ( t ) z + Z ( t , y ( t ) , z ( t ) , ∫ 0 t h 2 ( s , y ( s ) , z ( s ) , y ( s − τ ) , z ( s − τ ) ) d s ) (15)

where x = ( y , z ) T , τ ≥ 0 is a constant, initial condition is:

x ( t ) = φ ( t ) , t 0 − τ ≤ t ≤ t 0 ,

The homogeneous system of (15) is

Theorem: If (15) satisfies the following conditions:

1)

2)

3)

where

then the trivial solution of system (15) has uniform exponential Lipschitz asymptotic stability with respect to y, when the trivial solution of system (15)* has uniform stability and exponential asymptotic stability with respect to y.

Proof The V-Ляпунов function of (15)*, which is obtained under the condition of theorem, satisfies following conditions:

, (), (17)

for

Derivative the V-Ляпунов function

where

here

From condition of theorem and (17), when

By the first inequality of (16), the above can be expressed as follow:

then there exists

here select the appropriate small constant r such that

hence by the lemma [

For any solution of (15), from the inequality (18) and the first inequality of (16) we obtain

According to the proof of the theorem in [

In this paper, we use the method of differential inequalities with time-delay and integral inequalities to establish double stability criteria. As a result, studying the partial stability of differential equations becomes more important. In addition, the partial stability of differential equations is widely used in science and technology.

The authors are grateful to Professor Si Ligeng and the referee for several helpful comments.

Supported by Inner Mongolia Autonomous Region Higher Education Research Project (No.NJZY17064, NJZY16141).

The authors declare no conflicts of interest regarding the publication of this paper.

Huo, R. and Wang, X.L. (2019) Double Lipschitz Stability for Nonlinearly Perturbed Differential Systems with Multiple Delay. Journal of Applied Mathematics and Physics, 7, 3003-3011. https://doi.org/10.4236/jamp.2019.712210