ℎ-Stability of Linear Matrix Differential Systems

Abstract

This paper investigates the stability problem of linear matrix differential systems and gives some sufficient conditions of ℎ-stability for linear matrix system and its associated perturbed system by using the Kronecker product of matrices. An example is also worked out to illustrate our results.

1. Introduction

The theory of stability in the sense of Lyapunov is well known and is used in the real world. It is obvious 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 work with nonexponential types of stability. In recent years, Medina and Pinto [1, 2] extended the study of exponential stability to a variety of reasonable systems called ℎ-systems. They 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 the uniform Lipschitz stability) under some perturbations. Choi et al. [3] investigated ℎ-stability for the nonlinear differential systems by employing the notion of 𝑡_∞-similarity and the Lyapunov functions. And then, Choi et al. [4–6] also characterized the ℎ-stability in variation for nonlinear difference systems via 𝑛_∞-similarity and the Lyapunov functions and obtained some results related to stability for the perturbations of nonlinear difference systems.

However, as far as the author's scope, there are few discussions and results for matrix differential systems. In this paper, we shall investigate the ℎ-stability problem for linear matrix differential systems by employing the Kronecker product of matrices which can be found in Lakshmikantham and Deo's monograph [7]. Some preliminaries are presented in Section 2. A theorem is given in this section, which is important to complete the main results of this paper. In Section 3, sufficient conditions for the ℎ-stability are given for linear matrix system and its associated perturbed system. An example is also worked out at the end of this paper.

2. Preliminaries

Consider the linear matrix differential equation 𝑋^′=𝐴(𝑡)𝑋+𝑋𝐵(𝑡),𝑋𝑡₀=𝑋₀ (1) and its associated perturbed system 𝑌^′=𝐴(𝑡)𝑌+𝑌𝐵(𝑡)+𝑅(𝑡,𝑌),𝑌𝑡₀=𝑋₀, (2) where 𝐴,𝐵∈𝐶[𝑅⁺,𝑅^𝑛×𝑛], 𝑅∈𝐶[𝑅⁺×𝑅^𝑛×𝑛,𝑅^𝑛×𝑛], 𝑅(𝑡,0)≡0, and 𝑋,𝑌∈𝑅^𝑛×𝑛.

Now, we introduce the vec(⋅) operator which maps an 𝑚×𝑛 matrix 𝑃=(𝑝_𝑖𝑗) onto the vector composed of the rows of 𝑃vec(𝑃)=𝑝₁₁,…,𝑝_1𝑛,𝑝₂₁,…,𝑝_2𝑛,…,𝑝_𝑚1,…,𝑝_𝑚𝑛^𝑇. (3) Let us begin by defining the Kronecker product of matrices.

Definition 1 (see [7]). If 𝑃∈𝑅^𝑐×𝑑, 𝑄∈𝑅^𝑚×𝑛, then the Kronecker product of 𝑃 and 𝑄, 𝑃⊗𝑄∈𝑅^{𝑐𝑚×𝑑𝑛}, is defined by the matrix 𝑝𝑃⊗𝑄=¹¹𝑄𝑝₁₂𝑄⋯𝑝_1𝑑𝑄𝑝₂₁𝑄𝑝₂₂𝑄⋯𝑝_2𝑑𝑄𝑝........_𝑐1𝑄𝑝_𝑐2𝑄⋯𝑝_𝑐𝑑𝑄. (4)

Among the main properties of this product presented in [8], we recall the following useful ones: (1)vec(𝑃𝑋𝑄)=(𝑃⊗𝑄^𝑇)vec(𝑋), (2)vec(𝑃𝑋+𝑋𝑄)=(𝑃⊗𝐼+𝐼⊗𝑄^𝑇)vec(𝑋),where 𝑃,𝑋,𝑄, and 𝐼∈𝑅^𝑛×𝑛??and??𝐼 is an identity matrix.

Then, the equivalent vector differential systems of (1) and (2) can be written as 𝑥^′=𝐴⊗𝐼+𝐼⊗𝐵^𝑇𝑥,𝑥𝑡₀=𝑥₀,𝑦^′=𝐴⊗𝐼+𝐼⊗𝐵^𝑇𝑦+𝑟𝑡,𝑦,𝑦𝑡₀=𝑥₀, (5)(6) where 𝑥=vec(𝑋), 𝑦=vec(𝑌), 𝑟=vec(𝑅), 𝑟∈𝐶(𝑅⁺×𝑅^𝑛2,𝑅^𝑛2), and 𝑟(𝑡,0)≡0.

In order to investigate ℎ-stability of linear matrix equation and its associated perturbed system, we need to consider the following systems and their properties. The techniques and results are similar to those of [7].

Consider the linear differential system 𝑥^′=𝑃(𝑡)𝑥,𝑥𝑡₀=𝑥₀, (7) where 𝑃 is an 𝑛×𝑛 continuous matrix and its perturbation 𝑦^′=𝑃(𝑡)𝑦+𝐹𝑡,𝑦,𝑦𝑡₀=𝑥₀, (8) where 𝐹∈𝐶[𝑅⁺×𝑅^𝑛,𝑅^𝑛]. Suppose that the solution 𝑥(𝑡,𝑡₀,𝑥₀) of (7) exists for all 𝑡≥𝑡₀. The fundamental matrix solution Φ(𝑡,𝑡₀,𝑥₀) of (7) is given by [7] Φ𝑡,𝑡₀,𝑥₀=𝜕𝑥𝑡,𝑡₀,𝑥₀𝜕𝑥₀ (9) and Φ(𝑡₀,𝑡₀,𝑥₀)=𝐼.

We are now in a position to give the Alekseev formula, which connects the solutions of (7) and (8).

Lemma 2 (see [7]). If 𝑥(𝑡,𝑡₀,𝑥₀) is the solution of (7) and exists for 𝑡≥𝑡₀, any solution 𝑦(𝑡,𝑡₀,𝑥₀) of (8), with 𝑦(𝑡₀)=𝑥₀, satisfies the integral equation 𝑦𝑡,𝑡₀,𝑥₀=𝑥𝑡,𝑡₀,𝑥₀+^𝑡_𝑡0Φ𝑡,𝑠,𝑦𝑠,𝑡₀,𝑥₀𝐹𝑠,𝑦𝑠,𝑡₀,𝑥₀𝑑𝑠, (10) for 𝑡≥𝑡₀, where Φ(𝑡,𝑡₀,𝑥₀)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀.

Lemma 3 (see [7]). Assume that 𝑥(𝑡,𝑡₀,𝑥₀) is the solution of (7) through (𝑡₀,𝑥₀), which exists for 𝑡≥𝑡₀, then 𝑥𝑡,𝑡₀,𝑥₀=¹₀Φ𝑡,𝑡₀,𝑠𝑥₀𝑑𝑠𝑥₀, (11) where Φ(𝑡,𝑡₀,𝑥₀)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀.

The following theorem gives an analog of the variation of parameters formula for the solution of (2).

Theorem 4. Assume that 𝑥(𝑡,𝑡₀,𝑥₀) is the solution of (5) for 𝑡≥𝑡₀, let 𝐺𝑡,𝑡₀,𝑥₀=𝐴⊗𝐼+𝐼⊗𝐵^𝑇. (12) Then one has the following.
(i) Φ(𝑡,𝑡₀,𝑥₀)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀ exists and is the fundamental matrix solution of the variational equation 𝜑^′=𝐺𝑡,𝑡₀,𝑥₀𝜑, (13) such that Φ(𝑡₀,𝑡₀,𝑥₀)=𝐼, and therefore Φ𝑡,𝑡₀,𝑥₀=𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀, (14) where 𝑊(𝑡,𝑡₀) and 𝑍(𝑡,𝑡₀) are solutions of 𝑊^′=𝐴(𝑡)𝑊,𝑊𝑡₀=𝐼,𝑍^′=𝑍𝐵(𝑡),𝑍𝑡₀=𝐼, (15)(16) respectively.
?(ii) Any solution of (2) satisfies the integral equation 𝑌𝑡,𝑡₀,𝑋₀=𝑋𝑡,𝑡₀,𝑋₀+^𝑡_𝑡0𝑊(𝑡,𝑠)𝑅𝑠,𝑌𝑠,𝑡₀,𝑋₀𝑍(𝑡,𝑠)𝑑𝑠, (17) for 𝑡≥𝑡₀.

Proof. (i) It is obvious that Φ(𝑡,𝑡₀,𝑥₀)=𝜕𝑥(𝑡,𝑡₀,𝑥₀)/𝜕𝑥₀ exists and is the fundamental matrix solution of the variational equation 𝜑^′=𝐺𝑡,𝑡₀,𝑥₀𝜑, (18) such that Φ(𝑡₀,𝑡₀,𝑥₀)=𝐼.
Furthermore, we get 𝜑^′=𝐺𝑡,𝑡₀,𝑥₀𝜑=𝐴⊗𝐼+𝐼⊗𝐵^𝑇𝜑, (19) with the initial value 𝜑𝑡₀,𝑡₀,𝑥₀=𝑒,𝑒=vec(𝐼), (20) which has the solution 𝜑𝑡,𝑡₀,𝑥₀=𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀𝑒, (21) where 𝑊 and 𝑍 are the solutions of (15) and (16), respectively, and 𝐼 is the 𝑛×𝑛 identity matrix.
Therefore, Φ𝑡,𝑡₀,𝑥₀=𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀. (22)
(ii) Employing Lemma 2 and substituting for Φ the right-hand side of (22), we get 𝑦𝑡,𝑡₀,𝑥₀=𝑥𝑡,𝑡₀,𝑥₀+^𝑡_𝑡0𝑊(𝑡,𝑠)⊗𝑍^𝑇(𝑡,𝑠)𝑟𝑠,𝑦𝑠,𝑡₀,𝑥₀𝑑𝑠 (23) for 𝑡≥𝑡₀, where 𝑦(𝑡,𝑡₀,𝑥₀) is any solution of (6).
Now, we define 𝑋(𝑡,𝑡₀,𝑋₀), 𝑌(𝑡,𝑡₀,𝑋₀), and 𝑅(𝑡,𝑌) by 𝑥=vec(𝑋), 𝑦=vec(𝑌), and 𝑟=vec(𝑅). Thus, we have that 𝑌𝑡,𝑡₀,𝑋₀=𝑋𝑡,𝑡₀,𝑋₀+^𝑡_𝑡0𝑊(𝑡,𝑠)𝑅𝑠,𝑌𝑠,𝑡₀,𝑋₀𝑍(𝑡,𝑠)𝑑𝑠, (24) for 𝑡≥𝑡₀, where 𝑋(𝑡,𝑡₀,𝑋₀) is the unique solution of (1) for 𝑡≥𝑡₀.
The proof is completed.

3. Main Results

We firstly give some notions.

Definition 5. A generalized matrix valued norm from 𝑅^𝑚×𝑛 to 𝑅⁺ is a mapping ‖⋅‖∶𝑅^𝑚×𝑛→𝑅⁺ such that (a)‖𝑋‖≥0, ‖𝑋‖=0 if and only if 𝑋=0, (b)‖𝜆𝑋‖=|𝜆|‖𝑋‖, 𝜆 is a constant, (c)‖𝑋+𝑌‖≤‖𝑋‖+‖𝑌‖.

Definition 6. The zero solution of (1) is said to be (hS)ℎ-stability if there exist 𝑐≥1, 𝛿>0, and a positive bounded continuous function ℎ on 𝑅⁺ such that 𝑋𝑡,𝑡₀,𝑋₀≤𝑐𝑋₀ℎ(𝑡)ℎ⁻¹𝑡₀, (25)?for 𝑡≥𝑡₀≥0 and ‖𝑋₀‖≤𝛿, ℎ⁻¹(𝑡₀)=1/ℎ(𝑡₀). (hSV)ℎ-stability in variation if there exist 𝑐₁,𝑐₂≥1,𝛿>0, and a positive bounded continuous function ℎ on 𝑅⁺ satisfying 𝑊𝑡,𝑡₀≤𝑐₁ℎ(𝑡)ℎ⁻¹𝑡₀,𝑍𝑡,𝑡₀≤𝑐₂ℎ(𝑡)ℎ⁻¹𝑡₀, (26)?provided ‖𝑋₀‖≤𝛿, where 𝑊(𝑡,𝑡₀) and 𝑍(𝑡,𝑡₀) are given in Theorem 4.

Lemma 7 (see [4]). The linear system 𝑥^′=𝐴(𝑡)𝑥,𝑥𝑡₀=𝑥₀, (27) is ℎ𝑆 if and only if there exist a constant 𝑐≥1 and a positive continuous bounded function ℎ defined on 𝑅⁺ such that for every 𝑥₀ in 𝑅^𝑛, Φ𝑡,𝑡₀,𝑥₀≤𝑐ℎ(𝑡)ℎ⁻¹𝑡₀ (28) for all 𝑡≥𝑡₀≥0, where 𝐴(𝑡) is an 𝑛×𝑛 continuous matrix and Φ(𝑡,𝑡₀,𝑥₀) is a fundamental matrix of (27).

Theorem 8. The solution 𝑋=0 of (1) is ℎ𝑆 if and only if the solution 𝑥=0 of (5) is ℎ𝑆.

Proof. Necessity. Since the solution 𝑋=0 of (1) is ℎ𝑆, there exist 𝑐≥1, 𝛿>0, and a positive bounded continuous function ℎ on 𝑅⁺ such that 𝑋𝑡,𝑡₀,𝑋₀≤𝑐𝑋₀ℎ(𝑡)ℎ⁻¹𝑡₀ (29) for every 𝑡≥𝑡₀≥0,??‖𝑋₀‖≤𝛿, where 𝑋(𝑡,𝑡₀,𝑋₀) is the solution of (1), satisfying 𝑋^′=𝐴(𝑡)𝑋+𝑋𝐵(𝑡),𝑋𝑡₀=𝑋₀, (30) then we obtain vec𝑋^′=vec(𝐴(𝑡)𝑋+𝑋𝐵(𝑡)). (31) It follows that [vec(𝑋)]^′=𝐴⊗𝐼+𝐼⊗𝐵^𝑇vec(𝑋), (32) Thus, 𝑥𝑡,𝑡₀,𝑥₀=vec𝑋𝑡,𝑡₀,𝑋₀=𝑋𝑡,𝑡₀,𝑋₀≤𝑐𝑥₀ℎ(𝑡)ℎ⁻¹𝑡₀. (33)
Sufficiency. It can be easily proved by the same method. The proof is completed.

Theorem 9. The solution 𝑋=0 of (1) is ℎ𝑆 if and only if there exist a constant 𝑐≥1 and a positive continuous bounded function ℎ defined on 𝑅⁺ such that for every 𝑋₀ in 𝑅^𝑛×𝑛, 𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀≤𝑐ℎ(𝑡)ℎ⁻¹𝑡₀ (34) for all 𝑡≥𝑡₀≥0.

Proof. Sufficiency. Following Lemma 3 and Theorem 4, we have 𝑥𝑡,𝑡₀,𝑥₀=¹₀Φ𝑡,𝑡₀,𝑠𝑥₀𝑑𝑠𝑥₀,Φ𝑡,𝑡₀,𝑥₀=𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀. (35) It follows that 𝑥𝑡,𝑡₀,𝑥₀=¹₀𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀𝑑𝑠𝑥₀. (36) Hence, 𝑥𝑡,𝑡₀,𝑥₀≤𝑥₀¹₀𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀𝑥𝑑𝑠≤𝑐₀ℎ(𝑡)ℎ⁻¹𝑡₀,𝑡≥𝑡₀. (37) Therefore, the solution 𝑥=0 of (5) is ℎ𝑆. By Theorem 8, it implies that the solution 𝑋=0 of (1) is ℎ𝑆.
Necessity. If the solution 𝑋=0 of (1) is ℎ𝑆, then the solution 𝑥=0 of (5) is ℎ𝑆 using Theorem 8. By Lemma 7, we have Φ𝑡,𝑡₀,𝑥₀≤𝑐ℎ(𝑡)ℎ⁻¹𝑡₀. (38) From Theorem 4, we obtain that Φ𝑡,𝑡₀,𝑥₀=𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀, (39) Thus, 𝑊𝑡,𝑡₀⊗𝑍^𝑇𝑡,𝑡₀≤𝑐ℎ(𝑡)ℎ⁻¹𝑡₀. (40) This completes the proof.

Corollary 10. If the zero solution of (1) is ℎ𝑆𝑉, then the zero solution of (1) is ℎ𝑆.

Next, we offer sufficient conditions for the ℎ-stability of linear matrix differential systems by using the Lyapunov functions.

Defining the Lyapunov functions 𝐷⁺𝑉₍₅₎(𝑡,𝑥)=lim_𝛿→01sup𝛿×𝑉𝑡+𝛿,𝑥+𝛿𝐴⊗𝐼+𝐼⊗𝐵^𝑇𝑥−𝑉(𝑡,𝑥), (41) for (𝑡,𝑥)∈𝑅⁺×𝑅^𝑛2 and for the solution 𝑥(𝑡)=𝑥(𝑡,𝑡₀,𝑥₀) of (5), 𝐷⁺𝑉(𝑡,𝑥)=lim_𝛿→01sup𝛿[𝑉(𝑡+𝛿,𝑥(𝑡+𝛿))−𝑉(𝑡,𝑥)]. (42) Then, it is well known that 𝐷⁺𝑉₍₅₎(𝑡,𝑥)=𝐷⁺𝑉(𝑡,𝑥), (43) if 𝑉(𝑡,𝑥) is the Lipschitzian in 𝑥 for each 𝑡∈𝑅⁺.

Theorem 11. Suppose that ℎ(𝑡) is a positive bonded continuously differentiable function on 𝑅⁺. Furthermore, assume that there exists a function 𝑉(𝑡,𝑥) satisfying the following properties: (i)𝑉∈𝐶(𝑅⁺×𝑅^𝑛2,𝑅⁺), and 𝑉(𝑡,𝑥) is Lipschitzian in 𝑥 for each 𝑡∈𝑅⁺, (ii)‖𝑥‖≤𝑉(𝑡,𝑥)≤𝑐‖𝑥‖, (𝑡,𝑥)∈𝑅⁺×𝑅^𝑛2, 𝑐≥1, (iii)𝐷⁺𝑉₍₅₎(𝑡,𝑥)≤ℎ^′(𝑡)ℎ⁻¹(𝑡)𝑉(𝑡,𝑥), (𝑡,𝑥)∈𝑅⁺×𝑅^𝑛2.Then, the solution 𝑋=0 of (1) is ℎ𝑆.

Proof. Let 𝑥(𝑡,𝑡₀,𝑥₀) be the solution of (5). As a consequence of (iii), we obtain 𝑉𝑡,𝑋𝑡,𝑡₀,𝑥₀≤𝑉𝑡₀,𝑥₀exp^𝑡_𝑡0ℎ^′(𝑠)ℎ(𝑠)𝑡𝑑𝑠=𝑉₀,𝑥₀ℎ(𝑡)ℎ⁻¹𝑡₀. (44) From the condition (ii), we have 𝑥𝑡,𝑡₀,𝑥₀≤𝑐𝑥₀ℎ(𝑡)ℎ⁻¹𝑡₀,𝑡≥𝑡₀≥0. (45) By Theorem 8, we can easily get that the solution 𝑋=0 of (1) is ℎ𝑆. The proof is completed.

Now, we examine the properties of the perturbed linear matrix differential system.

Lemma 12 (see [9]). Suppose that 𝑘(𝑡,𝑥)∈𝐶(𝑅⁺×𝑅^𝑛,𝑅^𝑛) is strictly increasing in 𝑥 for 𝑡≥𝑡₀≥0 with the property 𝑥(𝑡)−^𝑡_𝑡0𝑘(𝑠,𝑥(𝑡))𝑑𝑠≤𝑦(𝑡)−^𝑡_𝑡0𝑘𝑠,𝑦(𝑡)𝑑𝑠,𝑡≥𝑡₀≥0 (46) for 𝑥,𝑦∈𝐶([𝑡₀,∞),𝑅^𝑛). If 𝑥(𝑡₀)<𝑦(𝑡₀), then 𝑥(𝑡)<𝑦(𝑡) for all 𝑡≥𝑡₀≥0.

Theorem 13. Assume that 𝑋=0 of (1) is ℎ𝑆𝑉 with the nonincreasing function ℎ₁ and ℎ₂. Consider the scalar differential equation 𝑢^′=𝑐𝑙(𝑡,𝑢),𝑢𝑡₀=𝑢₀,𝑤ℎ𝑒𝑟𝑒𝑐≥1. (47) Suppose that ‖𝑅(𝑡,𝑌)‖≤𝑙(𝑡,‖𝑌‖), (48) where 𝑙∈𝐶(𝑅⁺×𝑅⁺,𝑅⁺) is strictly increasing in 𝑢 for each fixed 𝑡≥𝑡₀≥0 with 𝑙(𝑡,0)=0.
If 𝑢=0 is ℎ𝑆, then the solution 𝑌=0 of (2) is also ℎ𝑆, whenever 𝑢₀=𝑐‖𝑌₀‖.

Proof. By Theorem 4, the solutions of (1) and (2) with the same initial values are related by 𝑌𝑡,𝑡₀,𝑌₀=𝑋𝑡,𝑡₀,𝑌₀+^𝑡_𝑡0𝑊(𝑡,𝑠)𝑅𝑠,𝑌𝑠,𝑡₀,𝑌₀𝑍(𝑡,𝑠)𝑑𝑠. (49) Then, we have 𝑌𝑡,𝑡₀,𝑌₀≤𝑋𝑡,𝑡₀,𝑌₀+^𝑡_𝑡0‖𝑊(×𝑅𝑡,𝑠)‖𝑠,𝑌𝑠,𝑡₀,𝑌₀‖Z(𝑡,𝑠)‖𝑑𝑠. (50) From Corollary 10, it easily follows that 𝑌𝑡,𝑡₀,𝑌₀≤𝑐₁𝑌₀ℎ₁(𝑡)ℎ₁⁻¹𝑡₀+^𝑡_𝑡0𝑐₂𝑐₃ℎ₂(𝑡)ℎ₂⁻¹(𝑠)²×𝑅𝑠,𝑌𝑠,𝑡₀,𝑌₀𝑌𝑑𝑠≤𝑐₀+𝑐^𝑡_𝑡0𝑙(𝑠,‖𝑌(𝑠)‖)𝑑𝑠, (51) where 𝑐=max{𝑐₁,𝑐₂𝑐₃}. Since ℎ₁(𝑡) and ℎ₂(𝑡) are nonincreasing, we obtain 𝑌𝑡,𝑡₀,𝑌₀−𝑐^𝑡_𝑡0𝑙(‖𝑌(𝑠𝑌𝑠,)‖)𝑑𝑠≤𝑐₀=𝑢₀=𝑢(𝑡)−^𝑡_𝑡0𝑐𝑙(𝑠,𝑢(𝑠))𝑑𝑠. (52) By Lemma 12, we have ‖𝑌(𝑡)‖<𝑢(𝑡) for all 𝑡≥𝑡₀≥0. Since 𝑢=0 of (47) is ℎ𝑆, ‖𝑌(𝑡)‖<𝑢(𝑡)≤𝑐₄𝑢₀ℎ(𝑡)ℎ⁻¹𝑡₀=𝑐₄𝑐𝑌₀ℎ(𝑡)ℎ⁻¹𝑡₀=𝑀𝑌₀ℎ(𝑡)ℎ⁻¹𝑡₀,𝑐₄≥1,𝑀=𝑐₄𝑐≥1. (53) This completes the proof.

Theorem 14. Assume that(i)the zero solution of (1) is ℎ𝑆𝑉,(ii)‖𝑅(𝑡,𝑌(𝑡))‖≤𝛾(𝑡)‖𝑌(𝑡)‖ provided that 𝛾(𝑡)>0 and ∫^∞_𝑡0𝛾(𝑡)𝑑𝑡<∞ for 𝑡₀≥0. Then, the solution 𝑌=0 of (2) is ℎ𝑆.

Proof. By Theorem 4, the solutions of (1) and (2) with the same initial values are related by 𝑌𝑡,𝑡₀,𝑋₀=𝑋𝑡,𝑡₀,𝑋₀+^𝑡_𝑡0𝑊(𝑡,𝑠)𝑅𝑠,𝑌𝑠,𝑡₀,𝑋₀𝑍(𝑡,𝑠)𝑑𝑠. (54) The assumptions (i) and (ii) yield 𝑌𝑡,𝑡₀,𝑋₀≤𝑋𝑡,𝑡₀,𝑋₀+^𝑡_𝑡0‖𝑊(𝑡,𝑠)‖𝑅𝑠,𝑌𝑠,𝑡₀,𝑋₀×‖𝑍(𝑡,𝑠)‖𝑑𝑠≤𝑐𝑋₀ℎ(𝑡)ℎ⁻¹𝑡₀+^𝑡_𝑡0𝑐₁𝑐₂ℎ₁(𝑡)ℎ₁⁻¹(𝑠)²×𝛾(𝑠)𝑌𝑠,𝑡₀,𝑋₀𝑑𝑠. (55) Then, by Gronwall's inequality, we get 𝑌𝑡,𝑡₀,𝑋₀≤𝑐𝑋₀ℎ(𝑡)ℎ⁻¹𝑡₀×exp^∞_𝑡0𝑐₁𝑐₂ℎ₁(𝑡)ℎ₁⁻¹(𝑠)²𝛾(𝑠)𝑋𝑑𝑠≤𝑀₀ℎ(𝑡)ℎ⁻¹𝑡₀, (56) where 𝑀=max{𝑐exp(∫^∞_𝑡0𝑐₁𝑐₂[ℎ₁(𝑡)ℎ₁⁻¹(𝑠)]²𝛾(𝑠)𝑑𝑠)}.
The proof is completed.

4. Example

In this section, we give a simple but illustrative example. Consider the matrix differential equation 𝑋^′(𝑡)=𝑋(𝑡)(𝑡),𝑋(0)=.−110−2+𝑋−100−11001 (57) Then, we can obtain the following equations: 𝑊^′=(0)−110−2𝑊,𝑊=𝐼,𝑍^′=𝑍(0)−100−1,𝑍=𝐼. (58)(59) The solutions of (58) and (59) are 𝑊(𝑡)=𝑒^−𝑡𝑒^−𝑡−𝑒^−2𝑡0𝑒^−2𝑡(𝑡)=𝑒,𝑍^−𝑡00𝑒^−𝑡, (60) respectively.

Then, 𝑊(𝑡)⊗𝑍^𝑇(𝑡)=𝑒^−2𝑡0𝑒^−2𝑡−𝑒^−3𝑡00𝑒^−2𝑡0𝑒^−2𝑡−𝑒^−3𝑡00𝑒^−3𝑡0000𝑒^−3𝑡. (61) Thus, we have 𝑊(𝑡)⊗𝑍^𝑇(𝑡)≤𝑐ℎ(𝑡)ℎ⁻¹(0), (62) where ℎ(𝑡)=𝑒^−2𝑡, 𝑐=5,‖𝑊(𝑡)⊗𝑍^𝑇(𝑡)‖=4𝑒^−2𝑡, ‖𝐷‖=∑^𝑚,𝑛_𝑖,𝑗|𝑑_𝑖𝑗|, 𝐷∈𝑅^𝑚×𝑛, and 𝐼 is an identity matrix. So, from Theorem 9, we can conclude that the solution 𝑋=0 of (57) is ℎ𝑆.

Acknowledgments

The authors would like to thank the reviewers for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (10971045 and 11271106).