Abstract
This paper studies the stability problem of a grey neutral stochastic system with distributed delays. The main feature of a grey system is that the parameters of system are evaluated by grey numbers, and system has the strong background in engineering applications. However, the literature dealing with the stability problem for grey system seems to be scarce. The main aim of this paper is to fill the gap. Based on the Lyapunov stability theorem and Itô’s formula, especially, using the decomposition technique of the continuous matrix-covered sets of grey matrix, we propose several novel sufficient conditions, which ensure our considered grey system in mean-square exponential stability and almost surely exponential stability. Furthermore, two examples are provided to show the effectiveness of the obtained results.
1. Introduction
In various physical and engineering systems, time-delay frequently causes oscillation, bifurcation, or instability. Meanwhile, the influence of stochastic phenomenon also cannot be ignored. So, time-delay stochastic system has gained many scholars’ attention and interest (see [1–9]). As pointed out in [10, 11], if the differences between nearby argument values are reduced, but the number of summands is incremental, we will find distributed delays phenomenon. In addition, we can describe many practical systems by neutral differential equations, Hence, neutral stochastic system with distributed delays has been extensively investigated by many scholars (see [12–20]). For instance, in [12], stability analysis of neutral type neural networks has been studied. Authors have proposed several delay-dependent criteria to ensure the networks to have a balance point. In [13], by the Lyapunov-Krasovkii functional, authors have studied the global exponential stability of neutral type neural networks with distributed time delays, and authors have presented new sufficient conditions of the system.
It is well known that we cannot obtain parameters of systems accurately. Naturally, in order to reflect many real systems, we have to evaluate the parameters of systems. When we evaluate the parameters of system by grey numbers, the system will be described uncertainly and turn into grey system (see [21]). Recently, grey system has come to play the important role in many fields and can be widely applied to physical and engineering systems, such as population ecology, communication systems, electric power systems, and so on. Due to the strong background in engineering applications, the stability analysis for grey system has become a focused topic of theoretical as well as practical importance. But, little attention has been taken to the stability problem for grey system except some paper [21–25]. In [21], the problem of exponential stability for a grey stochastic system with distributed delays has been studied, and the authors have proposed the delay-dependent criteria, which ensure the system in p-moment exponential robust stability. In [22], the authors have investigated robust stability problem for a grey stochastic nonlinear system with distributed delays, and several exponential stability criteria have been derived in terms of LMIs.
Motivated by the above discussion, using a similar method of [24], we will address the stability problem of grey system here. The main work and contribution of this paper are given as follows. (1) Compared with [24], this paper studies exponential stability problem of grey nonlinear systems with distributed delays for the first time, and the obtained results are more effective and general. (2) Different from traditional methods [10, 14, 20], for the uncertain systems, combining with concepts and conclusions of the Grey Theory (see Definitions 1 and 2 and Lemma 3), we adopt the new decomposition technique of the continuous matrix-covered sets of grey matrix. Then, several sufficient conditions are derived to ensure the systems in mean-square exponential stability and almost surely exponential stability. (3) Since the obtained criteria are presented LMIs, it can be easily solved by the Matlab LMI Control Toolbox. Meanwhile, two simple examples to illustrate the effectiveness of the obtained criteria are provided and solved.
The notation in this paper is fairly standard. The superscript “” represents the transpose and denotes the Euclidean norm. The notation , where is real symmetric matrix, means that is positive definite. Let be a probability space with a filtration and be the family of all continuous -valued functions on and be the family of all -measurable bounded -valued random variables .
2. Preliminaries and Problem Formulation
In this section, we consider the following grey system:where , , and are interval matrices, and are called the continuous matrix-covered sets of grey matrices , , and .
For grey system (1), we make the following assumptions and definitions:(H1) is locally Lipschitz continuous and satisfies the linear growth condition.(H2)There are constants , , for arbitrary ; we have
Definition 1 (see [21]). Grey system (1) is mean square exponentially stable, if for all and arbitrary , , , there exist constants and , such that
Definition 2 (see [21]). Grey system (1) is almost surely exponentially robustly stable, if for all and arbitrary , , , we have To prove our main results, the following lemmas are very important (in particular, Lemma 3).
Lemma 3 (see [21]). If is a grey matrix, is a number-covered sets of ; then for arbitrary matrix , we have (1),(2),(3),where , , , and .
Lemma 4 (see [26]). For any real vectors and any matrix with appropriate dimensions, , , we have
Lemma 5 (see [27]). For a given matrix with , , the following conditions are equivalent:(1),(2), .
3. Main Results and Proofs
Several novel stability conditions are proposed to guarantee grey system (1) in the mean-square exponential stability and almost surely exponential stability. The main results are given in the following theorems and corollaries.
Theorem 6. System (1) is mean square exponentially stable, if and there exist symmetric matrices , , and scalars , , , , such thatwhere Then, for all , we havehere, and is the unique positive solution of the following equationand , .
Proof. Using Lemma 5, implies that For convenience, let Firstly, arbitrarily fix , and , , ; we introduce the following Lyapunov-Krasovskii functional:Using Itô’s differential formula, we haveBy applying Lemmas 3 and 4, it follows that From assumptions (H2), we can getThen, combining (18)–(27) and noting (7), we havewhere is defined in (13).
Now, let , and by (28), we can get Thus, it can be seen thatSet .
We have .
Since , and , then (12) has a unique solution .
By (30) and if (12) holds, we can obtain On the other hand, it is easy to seeThen, it follows from (31) and (32) thatNote that Then, we can deriveBy (33), and note that definition of , it is easy to obtainSo, we also haveBy (37) and noting , we can getThat is, (10) holds. This completes the proof of Theorem 6.
Corollary 7. System (1) is mean square exponentially stable, if and there exist symmetric matrices and scalars , , , , , such that where Then, for all , we havehere, and is the unique positive solution of the following equationand , .
Proof. Let , ; similar to the method used in Theorem 6, the proof of Corollary 7 is completed.
Remark 8. If the interval matrices , , and are replaced by deterministic matrices , , and , then grey system (1) will become the deterministic stochastic system: Let , , and , and similar to the method used in Corollary 7, the criterion of the deterministic system (44) can be obtained.
Corollary 9. System (44) is mean square exponentially stable, if and there exist symmetric matrices and scalars , , , , , such that where Then, for all , we havehere, and is the unique positive solution of the following equationand , .
Theorem 10. Under the conditions of Theorem 6, grey system (1) is almost surely exponential stability. That is, for all , we have where , , and satisfies (12).
Proof. Under the conditions of Theorem 6, system (1) is mean square exponentially stable, so we haveUsing Doob’s martingale inequality, Cauchy inequality, from (51), we haveThen, using Borel-Cantelli lemma, and similar to the method used in Theorem (see [23, 24]), we complete the proof of Theorem 10.
4. Numerical Examples
Example 1. Consider a grey neutral systems with distributed delays: Here, the lower bound and upper bound matrices of , , and areMoreover, Then, we haveBy the calculation procedure programmed in [24], we can obtain and optimize scalars , , , and , which satisfy the criteria conditions in Theorem 6, and make the Lyapunov exponent much smaller. Now, the computed allowable values of constants and the Lyapunov exponent are shown in Table 1.
According to Theorem 6, and from Table 1, it is obvious that grey system (53) is exponentially stable in mean square, although the given system is described uncertainly. Moreover, it should be pointed out that [24] merely studies the exponential stability of a linear grey stochastic system. But, in this example, we can see that our work is suited to more general nonlinear grey stochastic systems. In the meantime, Theorem 6 only needs four positive scalars.
Example 2. Consider the stochastic system (44) withFor this deterministic system, if Corollary 9 in this paper is used, and by the Matlab LMI Control Toolbox, we can know that the Lyapunov exponent of this example is 1.0296.
For a comparison with the results of other methods and ours, the Lyapunov exponent is shown in Table 2.
It can be seen from Table 2 that,the Lyapunov exponent obtained by Corollary 9 is larger. That is, the result obtained by Corollary 9 is less conservative than that in [7, 15] and is an improvement of those results. Therefore, the criterion in Corollary 9 is valid. In addition, it is worth noting that since the parameters of given system are deterministic, the criterion in [7, 15] cannot be applied to Example 1. But Corollary 9 can be generalized, and we can obtain Corollary 7 to solve Example 1. Meanwhile, in this example, Corollary 9 only seeks a positive define matrix.
5. Conclusion
This paper has focused on the exponential stability problem for a grey neutral stochastic system with distributed delays. By the Lyapunov stability theorem, Itô’s formula, and decomposition technique of the continuous matrix-covered sets of grey matrix, some novel stability criteria have been obtained, which ensure our grey system in the mean-square exponential stability and almost surely exponential stability. Finally, we provide two examples to show the effectiveness of the proposed method.
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The research reported here is supported by the National Natural Science Foundation of China (no. 11301009) and the Key Project of Natural Science Foundation of Educational Committee of Henan Province (no. 16B110001).