LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS
LIPSCHITZ AND ASYMPTOTIC STABILITY FOR PERTURBED NONLINEAR DIFFERENTIAL SYSTEMS
The Pure and Applied Mathematics. 2014. Feb, 21(1): 11-21
Copyright © 2014, Korean Society of Mathematical Education
• Received : July 18, 2013
• Accepted : November 25, 2013
• Published : February 28, 2014
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YOON HOE, GOO

Abstract
The present paper is concerned with the notions of Lipschitz and asymptotic stability for perturbed nonlinear differential system knowing the corresponding stability of nonlinear differential system. We investigate Lipschitz and asymtotic stability for perturbed nonlinear differential systems. The main tool used is integral inequalities of the Bihari-type, in special some consequences of an extension of Bihari's result to Pinto and Pachpatte, and all that sort of things.
Keywords
1. INTRODUCTION
The notion of uniformly Lipschitz stability (ULS) was introduced by Dannan and Elaydi [8] . For linear systems, the notions of uniformly Lipschitz stability and that of uniformly stability are equivalent. However, for nonlinear systems, the two notions are quite distinct. In fact, uniformly Lipschitz stability lies somewhere between uniformly stability on one side and the notions of asmptotic stability in variation of Brauer [4] and uniformly stability in variation of Brauer and Strauss [3] on the other side. Gonzalez and Pinto [9] proved theorems which relate the asymptotic behavior and boundedness of the solutions of nonlinear differential systems.
In this paper, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear differential systems. To do this we need some integral inequalities. The method incorporating integral inequalities takes an important place among the methods developed for the qualitative analysis of solutions to linear and nonlinear system of differential equations. In the presence the method of integral inequalities is as efficient as the direct Lyapunov’s method.
2. PRELIMINARIES
We consider the nonlinear nonautonomous differential system
PPT Slide
Lager Image
where
PPT Slide
Lager Image
PPT Slide
Lager Image
and
PPT Slide
Lager Image
is the Euclidean n -space. We assume that the Jacobian matrix f x = ∂ f / ∂ x exists and is continuous on
PPT Slide
Lager Image
and f ( t , 0) = 0. Also, consider the perturbed differential system of (2.1)
PPT Slide
Lager Image
where
PPT Slide
Lager Image
, g ( t , 0) = 0. For
PPT Slide
Lager Image
, let
PPT Slide
Lager Image
For an n × n matrix A , define the norm | A | of A by | A | = sup |x|≤1 | Ax |.
Let x ( t , t 0 , x 0 ) denote the unique solution of (2.1) with x ( t 0 , t 0 , x 0 ) = x 0 , existing on [ t 0 ,∞). Then we can consider the associated variational systems around the zero solution of (2.1) and around x ( t ), respectively,
PPT Slide
Lager Image
and
PPT Slide
Lager Image
The fundamental matrix Φ( t , t 0 , x 0 ) of (2.4) is given by
and Φ( t , t 0 , 0) is the fundamental matrix of (2.3).
Before giving further details, we give some of the main definitions that we need in the sequel [8] .
Definition 2.1. The system (2.1) (the zero solution x = 0 of (2.1)) is called
(S) stable if for any 𝜖 > 0 and t 0 ≥ 0, there exists 𝛿 = 𝛿( t 0 , 𝜖) > 0 such that if | x 0| < 𝛿, then | x ( t )| < 𝜖 for all t t 0 ≥ 0,
(US) uniformly stable if the 𝛿 in (S) is independent of the time t 0 ,
(ULS) uniformly Lipschitz stable if there exist M > 0 and 𝛿 > 0 such that | x ( t )| ≤ M | x 0 | whenever | x 0 | ≤ 𝛿 and t t 0 ≥ 0
(ULSV) uniformly Lipschitz stable in variation if there exist M > 0 and 𝛿 > 0 such that |Φ( t , t 0 , x 0 )| ≤ M for | x 0 | ≤ 𝛿 and t t 0 ≥ 0,
(EAS) exponentially asymptotically stable if there exist constants K > 0 , c > 0, and 𝛿 > 0 such that
|x(t)| ≤ K |x0|e-c(t-t0), 0 ≤ t0t
provided that | x 0 | < 𝛿,
(EASV) exponentially asymptotically stable in variation if there exist constants K > 0 and c > 0 such that
|Φ(t,t0,x0)| ≤ K e-c(t-t0), 0 ≤ t0t
provided that | x 0 | < ∞.
We give some related properties that we need in the sequel.
We need Alekseev formula to compare between the solutions of (2.1) and the solutions of perturbed nonlinear system
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and g ( t , 0) = 0. Let y ( t ) = y ( t , t 0 , y 0 ) denote the solution of (2.5) passing through the point ( t 0 , y 0 ) in
PPT Slide
Lager Image
×
PPT Slide
Lager Image
.
The following is a generalization to nonlinear system of the variation of constants formula due to Alekseev [1] .
Lemma 2.2. Let x and y be a solution of (
PPT Slide
Lager Image
. 1 ) and (
PPT Slide
Lager Image
. 5 ), respectively. If
PPT Slide
Lager Image
then for all t such that
PPT Slide
Lager Image
PPT Slide
Lager Image
Lemma 2.3 ( [7] ) . Let u , λ 1 , λ 2 ,
PPT Slide
Lager Image
w ( u ) be nondecreasing in u and
PPT Slide
Lager Image
w ( u ) ≤ w(
PPT Slide
Lager Image
) for some v > 0. If , for some c > 0,
PPT Slide
Lager Image
then
PPT Slide
Lager Image
where
PPT Slide
Lager Image
u > 0, u 0 > 0 W -1 ( u ) is the inverse of W ( u ) and
PPT Slide
Lager Image
Lemma 2.4 ( [10] ) . Let u, p, q,w, and r C (
PPT Slide
Lager Image
) and suppose that, for some c ≥ 0, we have
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
Lemma 2.5 ( [15] ) . Let u ( t ), f ( t ), and g ( t ) be real-valued nonnegative continuous functions defined on
PPT Slide
Lager Image
, for which the inequality
PPT Slide
Lager Image
holds, where u 0 is a nonnegative constant. Then,
PPT Slide
Lager Image
Lemma 2.6 ( [12] ) . Let u , λ 1 , λ 2 , λ 3 C (
PPT Slide
Lager Image
), w C ((0,∞)) and w ( u ) be nondecreasing in u, u w ( u ). Suppose that for some c > 0,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where W, W -1 are the same functions as in Lemma 2.3 and
PPT Slide
Lager Image
Lemma 2.7 ( [13] ) . Let u, p, q,w, r C (
PPT Slide
Lager Image
), w C ((0,∞)) and w ( u ) be nondecreasing in u. Suppose that for some c ≥ 0,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
Lemma 2.8 ( [14] ) . Let the following condition hold for functions u ( t ), v ( t ) ∈ C [[ t 0 ,∞)
PPT Slide
Lager Image
) and k ( t, u ) ∈ C [[ t 0 ,∞) ×
PPT Slide
Lager Image
,
PPT Slide
Lager Image
) :
PPT Slide
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t t 0 and k ( s , u ) is strictly increasing in u for each fixed s ≥ 0. If u ( t 0 ) < v ( t 0 ), then u ( t ) < v ( t ), t t 0 ≥ 0.
Lemma 2.9 ( [5] ) . Let u, λ 1 , λ 2 , λ 3 C (
PPT Slide
Lager Image
), w C ((0,∞)) and w ( u ) be nondecreasing in u. Suppose that for some c > 0,
PPT Slide
Lager Image
Then
PPT Slide
Lager Image
where
PPT Slide
Lager Image
u > 0, u 0 > 0, W -1 ( u ) is the inverse of W ( u ) and
PPT Slide
Lager Image
3. MAIN RESULTS
In this section, we investigate Lipschitz and asymptotic stability for solutions of the nonlinear perturbed differential systems.
Theorem 3.1. Assume that x = 0 of (
PPT Slide
Lager Image
. 1) is ULS. Let the following condition hold for (
PPT Slide
Lager Image
.
PPT Slide
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):
PPT Slide
Lager Image
where W ( t , u ) ∈ C (
PPT Slide
Lager Image
×
PPT Slide
Lager Image
,
PPT Slide
Lager Image
) is monotone nondecreasing in u with W ( t , 0) = 0. Suppose that u ( t ) is any solution of the scalar differential equation
PPT Slide
Lager Image
existing on
PPT Slide
Lager Image
such that m ( t 0 ) < u ( t 0 ). If u = 0 of (3.1) is ULS, then y = 0 of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) is also ULS whenever M | y 0 | < u 0 .
Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Using the variation of constants formula, we have
PPT Slide
Lager Image
where Φ( t , t 0 , y 0 ) is the fundemental matrix of (2.4). Since x = 0 of (2.1) is ULS, it is ULSV by Corollary 3.6 [5] . Thus there exist M > 0 and 𝛿 > 0 such that |Φ( t , t 0 , y 0 )| ≤ M for t t 0 ≥ 0. Therefore, by the assmption, we have
PPT Slide
Lager Image
Hence | y ( t )| < u ( t ) by Lemma 2.8. Since u = 0 of (3.1) is ULS, it easily follows that y = 0 of (2.2) is ULS.
Corollary 3.2. Assume that x = 0 of (
PPT Slide
Lager Image
. 1 ) is ULS. Consider the scalar differential equation
PPT Slide
Lager Image
where u 0 ≥ 1, K ≥ 1 and a , k C (
PPT Slide
Lager Image
) satisfy the conditions
( a )
PPT Slide
Lager Image
where
PPT Slide
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( b )
PPT Slide
Lager Image
Then y = 0 of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) is ULS.
Proof. Let u ( t ) = u ( t , t 0 , x 0 ) be any solution of (3.2). Then, by Lemma 2.5 , we have
PPT Slide
Lager Image
Hence u = 0 of (3.2) is ULS. This implies that the solution y = 0 of (2.2) is ULS by Theorem 3.1.
Remark 3.3. In Corollary 3.2, it is needed that b 1 = ∞. The condition W (∞) = ∞ is too strong and it represents situations which are not stable. For example, if w ( u ) = u 𝛼 , then only 𝛼 ≤ 1 satisfies W (∞) = ∞ and 𝛼 < 1 is not stable. See [18] .
Corollary 3.4. Assume that x = 0 of (
PPT Slide
Lager Image
. 1 ) is ULS. Consider the scalar differential equation
PPT Slide
Lager Image
where u 0 ≥ 1, K ≥ 1, u , w C (
PPT Slide
Lager Image
), w ( u ) be nondecreasing in u and
PPT Slide
Lager Image
w ( u )≤ w (
PPT Slide
Lager Image
) for some v > 0, and a , k C (
PPT Slide
Lager Image
) satisfy the conditions
( a )
PPT Slide
Lager Image
where
PPT Slide
Lager Image
( b )
PPT Slide
Lager Image
and a, k L 1 (
PPT Slide
Lager Image
). Then y = 0 of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) is ULS.
Proof. Let u ( t ) = u ( t , t 0 , x 0 ) be any solution of (3.3). Then, by Lemma 2.3, we have
PPT Slide
Lager Image
Hence u = 0 of (3.3) is ULS. By Theorem 3.1, the solution y = 0 of (2.2) is ULS.
Corollary 3.5. Assume that x = 0 of (
PPT Slide
Lager Image
. 1 ) is ULS. Consider the scalar differential equation
PPT Slide
Lager Image
where w C ((0,∞), w ( u ) is nondecreasing on u and u w ( u ), u 0 ≥ 1, K ≥ 1 and a, b, k C (
PPT Slide
Lager Image
) satisfy the conditions
( a )
PPT Slide
Lager Image
where
PPT Slide
Lager Image
( b )
PPT Slide
Lager Image
L 1 (
PPT Slide
Lager Image
). Then y = 0 of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) is ULS.
Proof. Let u ( t ) = u ( t , t 0 , x 0 ) be any solution of (3.4). Then, Lemma 2.6, we have
PPT Slide
Lager Image
Hence u = 0 of (3.4) is ULS, and so by Theorem 3.1, the solution y = 0 of (2.2) is ULS. □
Theorem 3.6. For the perturbed (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
), we asssume that
PPT Slide
Lager Image
where a, b, k C (
PPT Slide
Lager Image
), a, b, k L 1 (
PPT Slide
Lager Image
), w C ((0,∞), and w ( u ) is nondecreasing in u,u w ( u ), and
PPT Slide
Lager Image
w ( u ) ≤ w (
PPT Slide
Lager Image
) for some v > 0,
PPT Slide
Lager Image
where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) is ULS whenever the zero solution of (
PPT Slide
Lager Image
. 1 ) is ULSV.
Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Since x = 0 of (2.1) is ULSV, it is ULS by Theorem 3.3 [8] . Applying Lemma 2.2, we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )|| y 0 | -1 . Now an application of Lemma 2.6 yields
PPT Slide
Lager Image
Hence we have | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < 𝛿. This completes the proof. □
Theorem 3.7. For the perturbed (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
), we asssume that
PPT Slide
Lager Image
where a, b, k C (
PPT Slide
Lager Image
), a, b, k L 1 (
PPT Slide
Lager Image
), w C ((0,∞), and w ( u ) is nondecreasing in u,u w ( u ), and
PPT Slide
Lager Image
w ( u ) ≤ w (
PPT Slide
Lager Image
) for some v > 0,
PPT Slide
Lager Image
where M ( t 0 ) < ∞ and b 1 = ∞. Then the zero solution of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) is ULS whenever the zero solution of (
PPT Slide
Lager Image
. 1 ) is ULSV.
Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Using the nonlinear variation of constants formula and the ULSV condition of x = 0 of (2.1), we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )|| y 0 | -1 . Now an application of Lemma 2.7 yields
PPT Slide
Lager Image
Thus we have | y ( t )| ≤ M ( t 0 )| y 0 | for some M ( t 0 ) > 0 whenever | y 0 | < 𝛿, and so the proof is complete. □
Theorem 3.8. Let the solution x = 0 of (
PPT Slide
Lager Image
. 1 ) be EAS. Suppose that the perturbing term g ( t , y ) satisfies
PPT Slide
Lager Image
where 𝛼 > 0, a, b, k C (
PPT Slide
Lager Image
), a, b, k L 1 (
PPT Slide
Lager Image
), w ( u ) is nondecreasing in u, and
PPT Slide
Lager Image
w ( u ) ≤ w (
PPT Slide
Lager Image
) for some v > 0. If
PPT Slide
Lager Image
where c = | y 0 | Me 𝛼t0 , then all solutions of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) approch zero as t → ∞
Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Since the solution x = 0 of (2.1) is EAS, we have |Φ( t , t 0 , x 0 )| ≤ Me -𝛼(t-t0) for some M > 0 and c > 0(Theorem 2 [2] ). Using Lemma 2.2, we have
PPT Slide
Lager Image
since e 𝛼t is increasing. Set u ( t ) = | y ( t )| e 𝛼t . An application of Lemma 2.4 obtains
PPT Slide
Lager Image
The above estimation yields the desired result. □
Theorem 3.9. Let the solution x = 0 of (
PPT Slide
Lager Image
. 1 ) be EAS. Suppose that the perturbing term g ( t , y ) satisfies
PPT Slide
Lager Image
where 𝛼 > 0, a, b, k , w C (
PPT Slide
Lager Image
), a, b, k L 1 (
PPT Slide
Lager Image
) and w ( u ) is nondecreasing in u. If
PPT Slide
Lager Image
where c = M | y 0 | e 𝛼t0 , then all solutions of (
PPT Slide
Lager Image
.
PPT Slide
Lager Image
) approch zero as t → ∞
Proof. Let x ( t ) = x ( t , t 0 , y 0 ) and y ( t ) = y ( t , t 0 , y 0 ) be solutions of (2.1) and (2.2), respectively. Using Lemma 2.2 and the assmptions, we have
PPT Slide
Lager Image
Set u ( t ) = | y ( t )| e 𝛼t . Since w ( u ) is nondecreasing, an application of Lemma 2.9 obtains
PPT Slide
Lager Image
where c = M | y 0 | e 𝛼t0 . From the above estimation, we obtains the desired result. □
Acknowledgements
The author is very grateful for the referee’s valuable comments.
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