Double Lipschitz Stability for Nonlinearly Perturbed Differential Systems with Multiple Delay

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.

• Time-Delay
• Nonlinear Systems
• Double Stability
• Differential Inequality
• Integral Inequality

1. Introduction

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” [1], in which he established the foundation of the stability theory. When studying nonlinear systems, especially studying dynamic systems or control systems, we cannot study the stability of all variables because of the technology difficulties, the limitation of practical conditions, or it is not necessary to study all variables considering the actual need. As a result, studying the partial stability of differential equations becomes more important. In addition, the partial stability is widely used in science and technology. For instance, the absolute stability of famous Lurie adjusting systems can be changed into a problem of partial stability. In a word, it is of practical significance to study the partial stability of differential equations.

In 1986, Dannan and Elaydi ( [2] ) introduced a new notion of stability, which is called uniform Lipschitz stability (ULS), for systems of differential equations

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 ( [3] ) considers the system with time-delay

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 ( [4] ) investigated the problems of ULS, EAS and GEASV for the following various perturbed differential systems of the nonlinear differential system (1) and

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. ( [5] [6] ) considered the following system

{ 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 [2] - [6]. 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.

2. Preliminaries

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. [7] Consider the homogeneous system

{ 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. [8] Suppose x i ( t ) be nonnegative continuous on R + , for all t ≥ t 0 (3) is satisfied, if ∃ K = c o n s t the following inequality holds:

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, the following inequality:

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.

3. Main Results

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 ,

is an matrix, is an matrix, is an matrix, they are all continuous for and satisfy the condition of existence and uniqueness theorem.

The homogeneous system of (15) is

. (15)*

Theorem: If (15) satisfies the following conditions:

1).

2).

3).

where, are nonnegative and not monotone decreasing for “”, , and

, ,

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:

, , (16)

, (), (17)

for,.

Derivative the V-Ляпунов function along (15), we get

,

where

here the notation of inner product.

From condition of theorem and (17), when we have

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

then there exists such that when and , we get

,

here select the appropriate small constant r such that

and,

hence by the lemma [9] [10], there exists such that for all we have

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 [11], we get, hence we obtain that the trivial solution of system (15) has uniform stability and uniform exponential Lipschitz asymptotic stability with respect to y.

4. Conclusion

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.

Acknowledgements

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

Supported

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

Conflicts of Interest

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

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