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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 253647, 11 pages
http://dx.doi.org/10.1155/2013/253647
Research Article

Stability of Impulsive Differential Systems

1Institute of Mathematics and Computer Science, University of Latvia, Raiņa bulvāris 29, Riga, LV-1459, Latvia
2Faculty of Physics and Mathematics, University of Latvia, Zeļļu iela 8, Riga, LV-1002, Latvia

Received 12 July 2013; Accepted 9 November 2013

Academic Editor: István Györi

Copyright © 2013 Andrejs Reinfelds and Lelde Sermone. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The asymptotic phase property and reduction principle for stability of a trivial solution is generalized to the case of the noninvertible impulsive differential equations in Banach spaces whose linear parts split into two parts and satisfy the condition of separation.

1. Introduction

The reduction principle in the theory of stability for systems of autonomous differential equations for the first time was proved by Pliss [1]. For systems of nonautonomous differential equations it was extended by Aulbach [2]; see also Pötzsche [3]. The analogy of the reduction principle for differential equations in Banach spaces was proved by Lykova [4] and for nonautonomous difference equations in Banach spaces by Reinfelds and Janglajew [5]. Several works [6, 7] are devoted to different modifications and applications of the reduction principle. In this paper, we generalize the reduction principle to the case of the noninvertible impulsive differential equations in Banach spaces whose linear part split into two parts and satisfy the condition of separation.

2. The Statement of the Problem

Let and be Banach spaces. By and we mean the Banach spaces of bounded linear operators. Consider the following system of impulsive differential equations: where(i)the mappings and are locally integrable in the Bochner sense;(ii)the mappings and are locally integrable in the Bochner sense with respect to for fixed and , and in addition they satisfy the uniform Lipschitz conditions (iii)for , , and , the mappings and satisfy the uniform Lipschitz conditions (iv)the mappings are homeomorphisms;(v)the moments of impulse form a strictly increasing sequence where the limit point may be only .

Without loss of generality we assume that the system (1) has the equilibrium points , ,

Using the suitable bump function it is possible for the analysis of local stability of the trivial solution to reduce to investigation of the global stability of the trivial solution if the nonlinear terms of (1) are uniform Lipschitz with respect to time and with a sufficient small constant in a fixed radius tubular neighbourhood of the trivial solution.

For simplicity, we assume that the linear part of (1) is decoupled in two separate parts. In many cases, this can be reached via the so-called kinematic similarity transformation [8, 9]. More generally via kinematic similarity transformation, the linear system can be reduced to the same almost reducible system [10], a system with a diagonal part and a small nondiagonal part. However, the kinematic transformation can grow unboundedly as the nondiagonal part tends to zero.

Definition 1 (see [11, 12]). By the solution to an impulsive system one means a piecewise absolutely continuous mapping with discontinuities of the first kind at the points which for almost all satisfies system (1) and for satisfies the conditions of a “jump.”

Note that condition (v) together with the Lipschitz property with respect to and of the right-hand side ensures that there is a unique solution.

Let be the solution of system (1), where . At the break points the values for all solutions are taken at unless otherwise indicated. For short, we will use the notation .

Let and be the evolutionary operators of the impulsive linear differential equations and, respectively,

We assume that the operators and satisfy the condition of separation [7]: where is the constant of separation.

To prove the theorems and lemmas, we use integrals which include evolutionary operators in their integrands. That is why it is more useful to estimate not the evolutionary operators but the corresponding integrals. Doing so, on the one hand, the conditions of theorems and lemmas are released from unnecessary technical limitations and, on the other hand, we obtain the conditions that are close to the necessary conditions.

If , , , and , then . Consequently, the integral converges if the spectrum of the mapping is located to the left of the spectrum of the mapping and the spectra are separated by a vertical line in the complex plane.

Let be a set of mappings that are continuous for and have discontinuities of the first kind for .

The set is a Banach space with the norm are a closed subsets of .

3. Auxiliary Lemma

Lemma 2. Let and . Then the following estimations are valid: where is the solution of the impulsive differential equations satisfying the initial condition .

We remark that and . It follows that (12) has a unique backward solution if .

Proof. The solution of the impulsive system (12) for is Taking into account that and satisfy the uniform Lipschitz conditions and properties, the solution can be estimated by
Multiplying the solution by and integrating from to , we obtain
Multiplying by and summing for all with respect to , we obtain
Summing up we get that
From the last inequality, we get that
Now we estimate the difference taking into consideration the properties of , , and :
Multiplying the difference by and integrating from to , we obtain
Multiplying the difference by and summing for all with respect to , we obtain
Summing up, we get that
Applying the first result of Lemma 2, we get From the last inequality we easily obtain (11).

4. Existence of a Lipschitz Invariant Manifold

Theorem 3. If , then there exists a unique piecewise continuous mapping satisfying the following properties:(i) for ;(ii); (iii).

Proof. Consider in the functional equation where is the solution of the impulsive differential equation system (12) satisfying the initial condition .
Consider the operator defined by the formula If , then satisfies the equality Then It follows that .
Taking into account that and satisfy the uniform Lipschitz conditions, we get that We have that and is a contraction in , and therefore there is only one solution satisfying the functional equation .
In addition for Therefore for uniqueness of solutions we get for The theorem is proven.

5. Behaviour of Solutions in the Neighbourhood of an Invariant Manifold

Theorem 4. Let . Then the following estimation is valid:

The inequality characterizes the integral distance between an arbitrary solution and an invariant manifold.

Proof. For an arbitrary map , piecewise continuous from the right with points of discontinuity of the first type, we have the following relation:
Set . Then for we obtain
Let us note that The third countable can be simplified:
Next we obtain
Now we consider
Thus,
We introduce the expression . For , we obtain the estimation Multiplying by , integrating, and summing analogously as in auxiliary Lemma 2, we obtain the inequality

6. Asymptotic Phase Type Property

Theorem 5. Let . Then for every solution of the impulsive system (1) there is a such solution of the impulsive system (12) that for all the following estimation is fulfilled: where

Proof. The set of mappings is a Banach space with the norm respectively.
Consider the functional equation in Consider the operator defined by formula
We have the following estimation: Besides If then . We have that is a contraction and . It follows that there is only one solution satisfying the functional equation . In addition for Let where . It follows that is a solution of (12) and This completes the proof of the theorem.

7. Stability of the Impulsive Equations

We assume in addition that Note that in case and we have .

Theorem 6. Let and . Then the following estimation is valid:

Proof. Since we get
From Theorem 4 of behaviour of solutions, we get inequality (40). Then doing the integration and summing up, inequality (55) is obtained.

Definition 7. A trivial solution of impulsive equation (1) is integral stable if for all there exists a such that for all and and one has

Definition 8. A trivial solution of impulsive equation (1) is asymptotically integral stable if it is integral stable and if there exists a such that for all and one has

Theorem 9. Let and . The trivial solution of impulsive equation (1) is integral stable, asymptotically integral stable, or integral unstable if and only if the trivial solution of impulsive equation (12) is integral stable, asymptotically integral stable, or integral unstable.

Proof. Suppose that the trivial solution of the system (12) is integral stable. Then for every , there is a such that for all and we have
Let and where Then for we get Therefore
Suppose that the trivial solution of the system (12) is asymptotically integral stable. Then It follows that taking into account that
If the trivial solution of (12) is integral unstable, then the trivial solution of (1) is integral unstable.
If the trivial solution of (1) is integral stable or asymptotically integral stable, then the trivial solution of (12) is also integral stable or asymptotically integral stable.
Let the trivial solution of (1) be integral unstable; then the trivial solution of (12) is integral unstable. Otherwise as before it follows that the trivial solution of (1) is integral stable. We get a contraction. The theorem is proven.

Theorem 10. Assume that the estimates are satisfied for some . If and , then

Proof. From Theorem 4 of behaviour of solutions, we get inequality (40). Multiplying by and doing the integration and summing up, the inequality is obtained.
Then from inequality (40) for we get the estimation

Theorem 11. Let , , The trivial solution of impulsive equation (1) is stable, asymptotically stable, or unstable if and only if the trivial solution of impulsive equation (12) is stable, asymptotically stable, or unstable.

Proof. Suppose that the trivial solution of the system (12) is stable. Then for every , there is a such that for all and we have .
Let and where Then for we get Therefore
Suppose that the trivial solution of the system (12) is asymptotically stable. Then It follows that
If the trivial solution of (12) is unstable, then the trivial solution of (1) is unstable.
If the trivial solution of (1) is stable or asymptotically stable, then the trivial solution of (12) is also stable or asymptotically stable.
Let the trivial solution of (1) be unstable; then the trivial solution of (12) is unstable. Otherwise as before it follows that the trivial solution of (1) is stable. We get a contraction. The theorem is proven.

Remark 12. Let be uniformly continuous on and let improper integral converge. Then [13, page 32].

Remark 13. If we replace assumption (54) by the stronger one then for it is possible to prove that , where and . Further, if is sufficiently small, then using Gronwall's lemma for all the following estimation is valid: where .

Acknowledgment

This work was partially supported by the Grant no. 345/2012 of the Latvian Council of Science.

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