ℎ-Stability for Differential Systems Relative to Initial Time Difference

Abstract

This paper investigates the relationship between an unperturbed differential system and a perturbed differential system that have initial time difference. Notions of ℎ-stability for differential systems with initial time difference are introduced, and stability criteria are formulated by using variation of parameter techniques.

1. Introduction

It is well known that, in applications, asymptotic stability is more important than stability, because the desirable feature is to know the size of the region of asymptotic stability. However, when we study the asymptotic stability, it is not easy to deal with nonexponential types of stability. Pinto [1] introduced the notion of ℎ-stability with the intention of obtaining results about stability for a weakly stable system (at least, weaker than those given exponential stability and uniform Lipschitz stability) under some perturbations and developed the study of exponential stability to a variety of reasonable systems called ℎ-systems. Since then, Choi and Ryu [2], Choi et al. [3], and Choi and Koo [4] investigated ℎ-stability problem for the nonlinear differential systems respectively, and Choi et al. [5, 6] characterized the ℎ-stability in variation for nonlinear difference systems via 𝑛_∞-similarity and Lyapunov functions and obtained some relative results. For the detailed results of ℎ-stability of impulsive dynamic systems on time scale and others systems can be found in [7–10].

At present, the investigation of differential systems with initial time difference has attracted a lot of attention. This is mainly because of the fact that when considering initial value problems, it is impossible not to make errors in the starting time in dealing with real world phenomena, that is, the solutions of the unperturbed differential system may start at some initial time and the solutions of the perturbed systems may start at a different initial time. When we consider such a change of initial time for each solution, we need to deal with the problem of comparing between any two solutions which start at different times. At present, there are two methods to discuss the stability problem with initial time difference: one is the differential inequalities and comparison principle, and the other is the method of variation of parameters. For the pioneering works in this area we can refer to the papers [11, 12]. After that, there are many stability results for various of differential and difference systems; see [13–20]. However, the above results were obtained by using comparison principle and differential inequalities; there are few stability criteria by using the method of variation of parameters; see [21–24].

In this paper, we attempt to extend the notion of ℎ-stability to differential systems with initial time difference, namely, initial time difference ℎ-stability (ITDℎS) and then establish some stability criteria for such differential systems by using the method of variation of parameters. The remainder of this paper is organized in the following manner. Some preliminaries are presented in Section 2. The notions of ℎ-stability for differential systems with initial time difference are given in this section. In Section 3, several stability criteria are established. Finally, an example is added to illustrate the result obtained.

2. Preliminaries

Let 𝑅⁺=[0,+∞) and 𝑅^𝑛 denotes the 𝑛-dimensional Euclidean space with appropriate norm ‖⋅‖.

Consider the differential systems: 𝑥^′=𝑓(𝑡,𝑥),𝑥𝑡₀=𝑥₀,𝑡≥𝑡₀,𝑡₀∈𝑅⁺,𝑥^′=𝑓(𝑡,𝑥),𝑥𝜏₀=𝑦₀,𝑡≥𝜏₀,𝜏₀∈𝑅⁺ (1)(2) and the perturbed differential system of (2): 𝑦^′=𝐹𝑡,𝑦,𝑦𝜏₀=𝑦₀,𝑡≥𝜏₀,𝜏₀∈𝑅⁺, (3) where 𝑓,𝐹∈𝐶[𝑅⁺×𝑅^𝑛,𝑅^𝑛] are locally Lipschitzian and 𝑓 has continuous partial derivatives 𝜕𝑓/𝜕𝑥 on 𝑅⁺×𝑅^𝑛. The above assumptions imply the existence and uniqueness of solutions through (𝑡₀,𝑥₀) and (𝜏₀,𝑦₀). A special case of (3) is where 𝐹(𝑡,𝑦)=𝑓(𝑡,𝑦)+𝑅(𝑡,𝑦), 𝑅(𝑡,𝑦) is the perturbation term. Let 𝜂=𝜏₀−𝑡₀>0. Furthermore, suppose that 𝑥(𝑡,𝑡₀,𝑥₀) is the given solution with respect to which we shall study stability criteria.

Let us begin by defining the following notions.

Definition 1. The solution 𝑥(𝑡,𝜏₀,𝑦₀) of the system (2) through (𝜏₀,𝑦₀) is said to be initial time difference ℎ-stability (ITDℎS) with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀), where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of the system (1), if and only if there exist 𝑐≥1 and a positive bounded continuous function ℎ defined on 𝑅⁺ such that 𝑥𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀ (4) for 𝑡≥𝜏₀ and ℎ⁻¹(𝑡)=1/ℎ(𝑡).

Similarly, we can define initial time difference ℎ-stability (ITDℎS) with respect to the solution 𝑦(𝑡,𝜏₀,𝑦₀) of the system (3) through (𝜏₀,𝑦₀).

We are now in a position to give the Alekseev’s formula, which is an important tool in the subsequent discussion.

Lemma 2 (see [25]). If 𝑥(𝑡,𝜏₀,𝑦₀) is the solution of (2) and exists for 𝑡≥𝜏₀, any solution 𝑦(𝑡,𝜏₀,𝑦₀) of (3), with 𝑦(𝜏₀)=𝑦₀, satisfies the integral equation: 𝑦𝑡,𝜏₀,𝑦₀=𝑥𝑡,𝜏₀,𝑦₀+^𝑡_𝜏0Φ𝑡,𝑠,𝑦𝑠,𝜏₀,𝑦₀𝑅𝑠,𝑦𝑠,𝜏₀,𝑦₀𝑑𝑠 (5) for 𝑡≥𝜏₀, where Φ(𝑡,𝜏₀,𝑦₀)=𝜕𝑥(𝑡,𝜏₀,𝑦₀)/𝜕𝑦₀.

The following lemma will also be needed in our investigations.

Lemma 3 (see [25]). Assume that 𝑥(𝑡,𝑡₀,𝑥₀) is the solution of (1) through (𝑡₀,𝑥₀), which exists for 𝑡≥𝑡₀, and then 𝑥𝑡,𝑡₀,𝑥₀=¹₀Φ𝑡,𝑡₀,𝑠𝑥₀𝑑𝑠𝑥₀, (6) where Φ(𝑡,𝑡₀,𝑥₀)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀.

3. Stability Criteria

We shall present, in this section, the stability criteria for differential systems with initial time difference.

Theorem 4. Let 𝑥(𝑡,𝜏₀,𝑦₀) and 𝑥(𝑡−𝜂,𝑡₀,𝑥₀) be the solutions of (2) and (1) through (𝜏₀,𝑦₀) and (𝑡₀,𝑥₀), respectively, 𝑡≥𝜏₀. Assume that (i)𝑣(𝑡,𝜏₀,𝑣₀)=𝑥(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀), in which 𝑣₀=𝑦₀−𝑥₀; (ii) there exists a positive bounded continuously differentiable function ℎ(𝑡) on 𝑅⁺ such that liminf_𝛿→0−𝑣𝑡,𝜏₀,𝑣₀+𝑓^∼𝑡,𝑣,𝜂𝛿−𝑣𝑡,𝜏₀,𝑣₀𝛿≤ℎ^′(𝑡)ℎ⁻¹(𝑡)𝑣𝑡,𝜏₀,𝑣₀, (7)?where 𝑓^∼(𝑡,𝑣,𝜂)=𝑓(𝑡,𝑥(𝑡−𝜂,𝑡₀,𝑥₀)+𝑣(𝑡,𝜏₀,𝑣₀))−𝑓(𝑡,𝑥(𝑡−𝜂,𝑡₀,𝑥₀)); (iii)𝑓 is locally Lipschitzian in time such that 𝑓𝑡,𝑥𝑡−𝜂,𝑡₀,𝑥₀−𝑓𝑡−𝜂,𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝐿₁(𝑡)𝜂𝐿₂𝜏₀, (8)?where 𝐿₂(𝜏₀)=∫_𝜏^+∞₀ℎ⁻¹(𝑠)ℎ(𝜏₀)𝐿₁(𝑠)𝑑𝑠,𝐿₁(𝑠)∈𝐶[𝑅⁺,𝑅⁺].

Then the solution 𝑥(𝑡,𝜏₀,𝑦₀) of the system (2) is ITDℎS with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀).

Proof. Define 𝑧(𝑡)=‖𝑣(𝑡,𝜏₀,𝑣₀)‖ for 𝑡≥𝜏₀, and then 𝑧(𝜏₀)=‖𝑦₀−𝑥₀‖. Also, 𝑣^′𝑡,𝜏₀,𝑣₀=𝑥^′𝑡,𝜏₀,𝑦₀−𝑥^′𝑡−𝜂,𝑡₀,𝑥₀=𝑓𝑡,𝑣𝑡,𝜏₀,𝑣₀+𝑥𝑡−𝜂,𝑡₀,𝑥₀−𝑓𝑡−𝜂,𝑥𝑡−𝜂,𝑡₀,𝑥₀. (9) Using a Taylor approximation for 𝑣(𝑡,𝜏₀,𝑣₀) and the conditions (i) and (ii), we arrive at 𝐷₋𝑧(𝑡)=lim_𝛿→0𝑣inf𝑡,𝜏₀,𝑣₀+𝑣^′𝑡,𝜏₀,𝑣₀𝛿−𝑣𝑡,𝜏₀,𝑣₀𝛿≤ℎ^′(𝑡)ℎ⁻¹(𝑡)𝑧(𝑡)+𝐿₁(𝑡)𝜂𝐿₂𝜏₀. (10) And then, from (10), we have 𝑧(𝑡)≤ℎ(𝑡)ℎ⁻¹𝜏₀×𝑧𝜏₀+𝜂𝐿₂𝜏_0𝜏^+∞₀ℎ⁻¹(𝑠)ℎ𝜏₀𝐿₁(𝑠)𝑑𝑠. (11) Moreover, using the condition (iii), we obtain 𝑧(𝑡)≤ℎ(𝑡)ℎ⁻¹𝜏₀𝑦₀−𝑥₀+𝜂. (12) Then from (12), we get 𝑥𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀. (13) So by Definition 1 with 𝑐=1, the solution 𝑥(𝑡,𝜏₀,𝑦₀) of (2) is ITDℎS with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀). This completes the proof.

Remark 5. Set ℎ(𝑡)=𝑒^−𝛽0𝑡, and then we can obtain Theorem 3.4 in [8].

Theorem 6. Let 𝑦(𝑡,𝜏₀,𝑦₀) be the solution of (3) through (𝜏₀,𝑦₀). Assume that (i) the solution 𝑥(𝑡,𝜏₀,𝑦₀) of (2) is ITDℎS with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀) for 𝑡≥𝜏₀, where 𝑥(𝑡,𝑡₀,𝑥₀) is any solution of (1); (ii) there exist 𝑐≥1, 𝛼>0 and a positive bounded continuous function ℎ defined on 𝑅⁺ such that Φ𝑡,𝑠,𝑦(𝑠)≤𝑐ℎ(𝑡)ℎ⁻¹(𝑠),𝑅𝑠,𝑦(𝑠)≤𝑟(𝑠)𝑦(𝑠), (14)?provided that 𝑦(𝑠,𝜏₀,𝑦₀)≤𝛼,𝑟(𝑠)∈𝐶(𝑅⁺,𝑅⁺) and ∫_𝜏^+∞₀𝑟(𝑠)𝑑𝑠<+∞.

Then the solution 𝑦(𝑡,𝜏₀,𝑦₀) of (3) is ITDℎS with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀).

Proof. Define 𝑣(𝑡,𝜏₀,𝑣₀)=𝑥(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀) and 𝑧(𝑡)=‖𝑣(𝑡,𝜏₀,𝑣₀)‖, and then 𝑧(𝜏₀)=‖𝑦₀−𝑥₀‖. The condition (i) yields 𝑥𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀. (15) By Lemma 2, it follows that 𝑦𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀=𝑣𝑡,𝜏₀,𝑣₀+^𝑡_𝜏0Φ𝑡,𝑠,𝑦(𝑠)𝑅𝑠,𝑦(𝑠)𝑑𝑠. (16) Now taking the norms of both sides and using the triangle inequality, we have 𝑦𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝑧(𝑡)+^𝑡_𝜏0Φ𝑡,𝑠,𝑦(𝑠)𝑅𝑠,𝑦(𝑠)𝑑𝑠. (17) From (15), we obtain 𝑦𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀+^𝑡_𝜏0Φ𝑡,𝑠,𝑦(𝑠)𝑅𝑠,𝑦(𝑠)𝑑𝑠. (18) Setting 𝑀^∗(𝑡)=‖𝑦(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀)‖ and using the triangle inequality, we have 𝑀^∗(𝑡)≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀+^𝑡_𝜏0𝑐ℎ(𝑡)ℎ⁻¹(𝑠)𝑟(𝑠)𝑀^∗(𝑠)+𝑑𝑠_𝑡𝜏0𝑐ℎ(𝑡)ℎ⁻¹(𝑠)𝑟(𝑠)𝑥𝑠−𝜂,𝑡₀,𝑥₀𝑑𝑠. (19) By using Lemma 3 and the condition (ii), we obtain 𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝛼𝑐ℎ(𝑡)ℎ⁻¹𝜏₀,for𝑥₀≤𝛼. (20) Hence, 𝑀^∗(𝑡)≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀+^𝑡_𝜏0𝑐ℎ(𝑡)ℎ⁻¹(𝑠)𝑟(𝑠)𝑀^∗(𝑠)𝑑𝑠+𝑐²𝛼ℎ(𝑡)ℎ⁻¹𝜏₀^𝑡_𝜏0𝑟(𝑠)𝑑𝑠. (21) Then we have 𝑁^∗(𝑡)≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀+^𝑡_𝜏0𝑟^∗(𝑠)𝑁^∗(𝑠)𝑑𝑠+𝑐²𝛼𝑁₁𝜏₀, (22) where 𝑟^∗(𝑡)=𝑐𝑟(𝑡),𝑁^∗(𝑡)=ℎ⁻¹(𝑡)ℎ(𝜏₀)𝑀^∗(𝑡), and ∫_𝜏^+∞₀𝑟(𝑠)𝑑𝑠=𝑁₁(𝜏₀).
By Gronwall's inequality, one gets 𝑀^∗(𝑡)≤𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀+𝑐²𝛼𝑁₁𝜏₀×ℎ(𝑡)ℎ⁻¹𝜏₀𝑒^{𝑐𝑁1(𝜏0)}. (23) Moreover, set 𝑐₁𝑦₀−𝑥₀+𝜏₀−𝑡₀=𝑐𝑦₀−𝑥₀+𝜏₀−𝑡₀+𝑐²𝛼𝑁₁𝜏₀𝑒^{𝑐𝑁1(𝜏0)} (24) and 𝑐₁≥1, we get 𝑦𝑡,𝜏₀,𝑦₀−𝑥𝑡−𝜂,𝑡₀,𝑥₀≤𝑐₁𝑦₀−𝑥₀+𝜏₀−𝑡₀ℎ(𝑡)ℎ⁻¹𝜏₀. (25) From Definition 1, it follows that the solution of (3) is ITDℎS with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀). This completes the proof.

4. Example

Now, we shall illustrate Theorem 6 by a simple example. Consider the differential systems 𝑥^′=−𝑥,𝑥𝑡₀=𝑥₀,𝑡≥𝑡₀,𝑡₀∈𝑅⁺,𝑥^′=−𝑥,𝑥𝜏₀=𝑦₀,𝑡≥𝜏₀,𝜏₀∈𝑅⁺, (26)(27) and the perturbed differential system of (27): 𝑦^′1=−𝑦+𝑡²𝑦,𝑦𝜏₀=𝑦₀,𝑡≥𝜏₀,𝜏₀∈𝑅⁺. (28) Define 𝑧(𝑡)=𝑥(𝑡,𝜏₀,𝑦₀)−𝑥(𝑡−𝜂,𝑡₀,𝑥₀); by direct calculation, we have the solution of (27) given by 𝑥(𝑡,𝜏₀,𝑦₀)=𝑦₀𝑒^{−𝑡+𝜏0}, which exists for all 𝑡≥𝜏₀, and Φ(𝑡,𝜏₀,𝑦₀)=𝜕𝑥(𝑡,𝜏₀,𝑦₀)/𝜕𝑦₀=𝑒^{−𝑡+𝜏0},??Φ(𝜏₀,𝜏₀,𝑦₀)=𝐼. ‖𝑧(𝑡)‖≤(‖𝑦₀−𝑥₀‖+|𝜏₀−𝑡₀|)𝑒^{−𝑡+𝜏0}, and then the solution of system (27) is ITDℎS with respect to 𝑥(𝑡−𝜂,𝜏₀,𝑥₀).

Now, let us begin to consider the perturbation term 𝐹(𝑡,𝑦)=(1/𝑡²)𝑦 of (28), and we have ‖(1/𝑡²)𝑦‖≤(1/𝑡²)‖𝑦‖, where ∫_𝜏^+∞₀(1/𝑡²)𝑑𝑡<+∞. Then by Theorem 6, we can conclude that the solution 𝑦(𝑡,𝜏₀,𝑦₀) of (28) is ITDℎS with respect to the solution 𝑥(𝑡−𝜂,𝑡₀,𝑥₀).

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11271106) and the Natural Science Foundation of Hebei Province of China (A2013201232).