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

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)withoutthelineargrowthconditionandexaminestheboundednessandexponentialstability.Twoillustrativeexamplesaregiventoshowtheeffectivenessofourtheoreticalresults.


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][12][13][14][15][16][17][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.

Notations and Definitions
Throughout this paper, unless otherwise specified, we use the following notations. | ⋅ | denotes both the Euclidean vector norm in R and Frobenius matrix norm in R × . If is a vector or matrix, its transpose is denoted by . If is a matrix, its trace norm is denoted by | | = √ trace( ). Let (Ω, F, ) be a complete probability space with a filtration {F t } ≥0 satisfying the usual conditions; that is, it is right continuous and increasing while F 0 contains all -null sets. Let ( ) = ( 1 ( ), . . . , ( )) be a -dimensional Brownian motion defined on this probability space. Let R + = [0, ∞) and Let ( ) be a Markov chain (independent of ( )) taking values in a finite state space S = {1, 2, . . . , }. Assume the generator of ( ) is denoted by Γ = ( ) × , so that where > 0. Here is the transition rate from to and Let us consider the following -dimensional nonlinear HNSFDE: with initial data 0 = ∈ where and (0, ) = 0.
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.
where ( , ) = ( ( , ) 1 , . . . , ( , ) ) , In the following sections, we will impose the some conditions on the diffusion operator L for the global solution and its asymptotic behavior.

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

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 = 1 in (13).
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). 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 ( ).
Abstract and Applied Analysis 7

The Boundedness and Exponential Stability of HNSDDEs
In this section, we investigate the exponential stability of the hybrid NSDDE with varying delay where ( ) : R + → [0, ] is a continuously differentiable function such that ( ) ≤ (47) for some constant < 1, while For (46), we impose the following assumptions.
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 R × R × S to R by for each ( , ) ∈ 2 (R × S; R + ), and we will impose the same conditions on the diffusion operator for the global solution and its asymptotic behavior.
(ii) If, in addition, = 0, then the solution to (46) has properties that lim sup 0)) , (0)) , (0)) 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 8 Abstract and Applied Analysis of Theorem 3. By the generalized Itô formula (see [10]) and condition (53), we can obtain that, for any > 0 and ≥ 0, 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 → ∞.
If the delay ( ) = is a fixed constant, then = 0. Hybrid system (46) becomes the following HNSDDE: Resorting to Theorem 10, we have the following corollary.

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 S = {1, 2} with generator Γ = ( ) 2 × 2 = ( −2 2 1 −1 ) .
Assume that ( ) and ( ) are independent.