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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 138031, 12 pages

http://dx.doi.org/10.1155/2013/138031

## The Boundedness and Exponential Stability Criterions for Nonlinear Hybrid Neutral Stochastic Functional Differential Equations

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China

Received 26 April 2013; Accepted 25 June 2013

Academic Editor: Marcia Federson

Copyright © 2013 Xiaofeng Zong et al. 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

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results.

#### 1. Introduction

Many dynamic systems not only depend on the present and past states but also involve derivatives with delays. Neutral differential equations have been used to model such systems. Deterministic neutral differential equations were introduced by Hale and Meyer [1] and discussed in Hale and Lunel (see [2]) and Kolmanovskii et al. (for details see also [3, 4]), among others. Such equations were used to study two or more simple oscillatory systems with some interconnections between them, such as Brayton [5], Rubanik [6], and Driver [7].

Generally speaking, many practical systems commonly encounter stochastic perturbations and may experience abrupt changes in their structure and parameters caused by phenomena such as component failures or repairs and abrupt environmental disturbances. Of course, there is no exception to neutral systems, mentioned previous. Taking these stochastic factors into account, Mao and Yuan developed hybrid systems driven by Brownian motion and continuous-time Markovian chain to cope with such a situation (see [8]). Hu et al. [9] investigated the stability and boundedness of stochastic differential delay equations with Markovian switching. Kolmanovskii et al. [10] discussed the neutral stochastic delay differential equations with Markovian switching, also known as hybrid neutral stochastic delay differential equations (HNSDDEs).

The boundedness and stability analysis of the neutral stochastic systems without switching has attracted much attention; see [11–18] to mention a few. For hybrid neutral systems, studying boundedness and stability of the solutions is also a challenging and interesting work. Kolmanovskii et al. [10] established a fundamental theorem of HNSDDEs and discussed the boundedness and exponential stability of the solutions. They also gave an example to show that Markovian can average the subsystems; that is, when some subsystems are stable and others are not stable, the overall system formed by the Markovian switching may be stable. Bao et al. [19] discussed stability in distribution of the HNSDDEs. Hu and Wang [20] studied the stability in distribution for the general HNSFDEs. The stability of HNSDDEs with interval uncertainty was investigated in [21]. Mao et al. [22] gave a criterion related to almost surely asymptotic stability of HNSDDEs. These results are undoubtedly remarkable.

However, there are few publications on the boundedness and exponential stability of the general HNSFDEs with highly nonlinear terms. To fill in this gap, this work gives the boundedness and exponential stability criterions for such HNSFDEs. Moreover, when this HNSFDE degenerates to the HNSDDE, our stability criterions improve the related results in [10]. Further, these stability criterions can also be used to investigate the exponential stability of NSFDEs or NSDDEs with more accurate Lyapunov exponent bound than that obtained in [23, 24].

The rest of the paper is arranged as follows. The next section provides necessary notations and definitions for the use of this paper. Section 3 establishes the boundedness and exponential stability criterions of the solutions to HNSFDEs. Section 4 further gives the generalized results for the HNSDDEs with variable time delay. Finally, two illustrative examples are provided to show the effectiveness of our theoretical results.

#### 2. Notations and Definitions

Throughout this paper, unless otherwise specified, we use the following notations. denotes both the Euclidean vector norm in and Frobenius matrix norm in . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by . Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and increasing while contains all -null sets. Let be a -dimensional Brownian motion defined on this probability space. Let and . Denote by the family of continuous functions from to with the norm . Let be the family of all -measurable bounded valued random variables . represents , and denotes .

Let be a Markov chain (independent of ) taking values in a finite state space . Assume the generator of is denoted by , so that where . Here is the transition rate from to and if while . Let us consider the following -dimensional nonlinear HNSFDE: with initial data and , where , , and . In order to guarantee the existence and uniqueness of the solution to (2), we give the following assumptions for the functionals , , and .

*Assumption 1 (local Lipschitz condition). *and satisfy the local Lipschitz condition; that is, for each there exists a positive constant such that for any maps , with
where .

*Assumption 2 (contractive mapping). *There exists a positive constant such that for all , and
and .

Note that the previous assumptions are standard for the existence and uniqueness of the local solutions (see [19, 22]). Additional conditions should be imposed for the local solution to be global. In view of this, we need a few more notations. Let denote the family of all nonnegative functions on which are continuously twice differentiable in . For each , define an operator from to : where

In the following sections, we will impose the some conditions on the diffusion operator for the global solution and its asymptotic behavior.

#### 3. The Boundedness and Exponential Stability of HNSFDEs

The following theorem gives the boundedness and exponential stability criterions of the solution to (2).

Theorem 3. *Let Assumptions 1 and 2 hold. Assume that there are two functions , , three probability measures , , on , and a number of positive constants , , , , , , , , such that for any and **
If and , then we have the following assertions:*(i)* for any given initial data , there is a unique global solution to the hybrid system (2) on ; *(ii)* the solution obeys
where with and defined by
and for sufficiently small . *(iii)* If, in addition, , then the solution of (2) has properties that
where . *

*Proof. *We prove these three assertions, separately. For any given initial data , by Assumptions 1 and 2, there exists a unique maximal local strong solution to (2) on , where is the explosion time. To show that this solution is global, we only need to prove that a.s. Define , then by Assumption 2, we have
Let be sufficiently lager positive number, such that . For each , define the stopping time . Clearly, is increasing as and a.s. If we can show a.s., then , which implies that the solution is actually global. By the generalized Itô formula (see [20]) and condition (10), we can obtain that, for any and ,
By the Fubini theorem, we compute
Similarly,
Substituting (18) and (19) into (17) yields
where
Note that

Hence,

Then, for any ,
which together with the arbitrariness of implies that a.s. Therefore, the solution is global, and assertion (i) follows.

Then by Itô’s formula and condition (10), we have, for any ,
For and any , we have
where we used the Hölder inequality and condition (8). Substituting (26) into (25), we obtain

Define a probability measure on

then from (27), we have
By the Fubini theorem
we have from (29)
Denote
Let be fixed; then it is easy to obtain and , which implies that for any fixed , function has a unique positive root, denoted by . Choose a such that
Noting that for any , and , we therefore have
for some positive constant . Letting , we have
which implies the desired assertion (11). Assertion (12) can be obtained from (20) by letting . Hence assertion (ii) follows.

Let . For any , we have that

By (34) and (36), we have for
This inequality also holds for all . In view of , there exists a positive number such that
Therefore,

which implies
Finally, the required inequality (14) follows by taking logarithm and limitation. Inequality (15) can be also obtained from (20) by letting . Hence assertion (iii) follows.

*Remark 4. *If the Markovian switching vanishes, Theorem 3 is also true and gives the th moment exponential stability with the decay rate bigger than that in [24, Theorem 2]. Since the decay rate in [24] is the special case of Theorem 3 with in (13).

If , then we directly obtain the following corollary.

Corollary 5. *Let Assumptions 1 and 2 hold. Assume that there exist a function , two probability measures , on , and a number of positive constants , , , , , such that for any and **
If , then for any given initial data , the solution of (2), denoted by , has property that
**
where and satisfy
**
and for sufficiently small .*

Although the th moment exponential stability and almost sure exponential stability of the exact solution do not imply each other in general, under a restrictive condition the th moment exponential stability implies almost sure exponential stability (cf. [11]). Here, we give the following theorem about the almost sure exponential stability of the exact solution to (2).

Theorem 6. *Let . Assume that there exists a constant such that
**
Then (42) implies
**
In other words, the pth moment exponential stability implies almost sure exponential stability.*

*Remark 7. *One may question that whether the semimartingale technique can be used to obtain the almost sure exponential stability directly. In fact, semimartingale technique may fail, since it may not be true to transfer the almost sure exponential stability from to .

#### 4. The Boundedness and Exponential Stability of HNSDDEs

In this section, we investigate the exponential stability of the hybrid NSDDE with varying delay where is a continuously differentiable function such that for some constant , while

For (46), we impose the following assumptions.

*Assumption 8 (local Lipschitz condition). * and satisfy the local Lipschitz condition; that is, for each there exists a positive constant such that
for all and with .

*Assumption 9 (contractive mapping). * is a contractive mapping; that is, there exists a positive constant such that for all , and

Under the previous two assumptions, HNSDDE (46) admits a unique local solution. We also need more conditions to guarantee that the local solution is actually global. So we introduce an operator from to by for each , and we will impose the same conditions on the diffusion operator for the global solution and its asymptotic behavior.

Although HNSDDE can be regarded as the special case of HNSFDEs, we still establish the boundedness and exponential stability criterions of the solution for (46) so as to obtain more accurate results.

Theorem 10. *Let Assumptions 8 and 9 hold. Assume that there are functions , as well as a number of positive constants , , , , , , , such that for any , and ,
**
If and , then for any given initial data , (46) admits a unique global solution . Moreover, we have the following assertions:*(i)*the solution obeys
where with and defined by
and for sufficiently small . *(ii)*If, in addition, , then the solution to (46) has properties that
*

*Proof. *The proof is similar to that of Theorem 3, so we only give an outlined one. Denote . Let be the stopping time defined similarly in the proof of Theorem 3. By the generalized Itô formula (see [10]) and condition (53), we can obtain that, for any and ,

Noting that
then we have
Then by the similar arguments used in the proof of Theorem 3, we easily obtain as ; that is, the solution is global. The desired assertions (55) and (58) follow from (61) by letting .

Applying It’s formula to and using condition (53), we have for any
For and any , we have
where we used the Hlder inequality and Assumption 9. Substituting (63) into (62), we therefore obtain
Noting that
then we have from (64)
where
Let be fixed; then it is easy to obtain and , which implies that for any fixed function has a unique positive root denoted by . Choose a such that
Noting that for any , and . We therefore have
for some positive constant . By the similar skills used in the proof of Theorem 3, we can easily obtain the desired assertions (54) and (57).

If the delay is a fixed constant, then . Hybrid system (46) becomes the following HNSDDE: Resorting to Theorem 10, we have the following corollary.

Corollary 11. *Let Assumptions 8 and 9 hold. Assume that there are functions , as well as a number of positive constants , , , , , , , such that for any and **
If and , then for any given initial data , (70) admits a unique global solution . Moreover, we have the following assertions: *(i)*the solution obeys
where with and defined by
and for sufficiently small . *(ii)*If, in addition, , then the solution of (70) has properties that
*

Further, if , Corollary 11 implies.

Corollary 12. *Let Assumptions 8 and 9 hold. Assume that there is a function and a number of positive constants , , , , such that for any and **
If , then for any given initial data , (70) admits a unique global solution . Moreover, the solution obeys
**
where with and defined by
**
and for sufficiently small .*

*Remark 13. *Corollary 12 improves Theorem 5.2 in [10, Chap. 5, pp. 838]. In [10], Theorem 5.2 states that if the assumptions and conditions in Corollary 12 hold, then
where
with being the unique root to equation
It is easy to see . Moreover, . That means for sufficiently small , where is defined in Corollary 12.

#### 5. Examples

In this section, we give an example to illustrate the usefulness and flexibility of the theorems developed previously. Let be a scalar Brownian motion. Let be a right-continuous Markov chain value in with generator Assume that and are independent.

*Example 1. *Consider the one-dimensional linear HNSDDEs
where , , , , , , and . To find out whether (83) is mean-square exponential stability, we use the Lyapunov function
where and . One can show that
By the elementary inequalities , we have
Similarly,
Let , , , and . Then we have from (86), and (87) for each ,
Then one can compute by Theorem 3. If , then
If , the solution to HNSDDE (83) has the property

*Example 2. *Consider the HNSDDE:
where the functions ,