Fractional-Order Systems: Control Theory and ApplicationsView this Special Issue
-Stability in -th Moment of Neutral Impulsive Stochastic Functional Differential Equations with Markovian Switching
In this paper, we investigate the -stability in q-th moment for neutral impulsive stochastic functional differential equations with Markovian switching (NISFDEwMS). Moreover, -stability in q-th moment is studied by using the Lyapunov techniques and a new Razumikhin-type theorem to prove our result. Finally, we check the main result by a numerical example.
The aim of stochastic differential equations is to provide a mathematical model for a differential equation disturbed by a random noise. Consider an ordinary differential equation of the form
An ordinary differential equation is used to describe the evolution of a physical system. If we take into account the random disturbances, we add a noise term, which will be of the form , where denotes a Brownian motion and , for this instant, is a constant which corresponds to the intensity of the noise. Then, we have a stochastic differential equation (SDE) of the form
Moreover, we have the following integral form:
We generalize equation (3) by taking depending on the state of the system to an instant :
In view of the integral in , this equation will depend on the theory of stochastic integral (see ).
One of the most important qualitative studies of SDE is the stability theory.
A special case of SDE is the neutral stochastic functional differential equations (NSFDE).
In the literature, there are many research papers on the stability theory of NSFDE (see [4–6]). Recently, stability of SFDE with Markovian switching (SFDEwMS) has received a lot of attention (see [7–11]).
To the best of our knowledge, there are a few papers on the -stability or stability of SDE with general decay rate. In [14–16], the authors established the stability with general decay rate of stochastic functional differential equations with finite and infinite delay.
In the literature, there is no existing paper on the -stability of NISFDEwMS.
In this paper, we study the -stability in -th moment of NISFDEwMS. In view of the other related topics on NISFDEwMS, the main contributions of this paper are to obtain a new technique to study the -stability in a hybrid impulsive neutral stochastic system, to develop a new Razumikhin-type theorem on the Lyapunov function to prove the -stability of the system, and to propose new theories on asymptotic properties in -th moment of NISFDEwMS. In this sense, our results generalize the paper  in the case of NISFDEwMS.
The paper is organized as follows. In Section 2, we cite some definitions and basic notions. In Section 3, we investigate the -stability in -th moment of NISFDEwMS. In Section 4, one numerical example is given to show the interest of our main results.
2. Preliminaries and Definitions
The complete probability space is denoted by , where is a filtration satisfying the usual conditions. is an -dimensional Brownian motion defined on the probability space. Let and be the family of continuous functions from to with the norm , where for any . If is a matrix, its trace norm is denoted by , where its norm is given by .
Let , where exists and for all but at most a finite number of points t, with the uniform norm , where and denote the right-hand and left-hand limits of at , respectively. For , let be the family of all function such that .
Let be a right-continuous Markov chain on the probability space taking values in a finite state space with as a generator given bywhile . Here, is the transition rate from to , if , while
We consider the following NISFDEwMS:where is defined by , which is a -valued stochastic process, are impulsive moments satisfying and , and represents the jump in the state at with determining the size of the jump. Furthermore, we assume that
Let be the family of all nonnegative functions on , which are twice continuously differentiable with respect to and once continuously differentiable with respect to .
Define the operator by (see )where
Throughout this paper, we assume that , , , and are smooth enough to ensure the global existence and uniqueness of solutions for all . For any , there is a unique solution satisfying (7), which is continuous on the left-hand side and have a finite limit on the right-hand side. We assume that , , and , where , , and , which implies that 0 is an equilibrium solution.
Definition 1. The function is said to be -type function if it satisfies the following conditions:(i)It is continuous and nondecreasing in and continuously differentiable in (ii), , and (iii)For any , ,
Definition 2. System (7) is said to be -stable in -th moment (-s.q.m), if there exists a function satisfying Definition 1 and positive constants and such that, for any ,which is equivalent towhen , it is said to be -stable in mean square.
3. Main Results
Theorem 1. Assume that , constants , , , , , , , , , , and , and a function satisfying Definition 1 such that(i) and .(ii) and satisfying where and those satisfying(iii) satisfying(iv).(v)For any , .
Then, the solution of system (7) is -stable in -th moment, where and .
Proof. For any , let be the solution of (7) and , where . Let be sufficiently small such that . By using Itô formula, we obtainLet , we can derive that, for ,whereLet , then we have for ,By (iii), we obtainLet such thatWe can claim that, for ,By (19), it is clear to see that for . We will prove thatOtherwise, there exists satisfying and , for . By the continuity of in , there exists satisfyingFor and , we obtainThus, we haveCombining (26) with (19) and (ii), for , we haveThen,This is a contradiction. Therefore, (23) holds. By (iii), we can deriveNow, we will prove thatSuppose that (30) does not hold, there exists such thatBy the continuity of in , there exists satisfyingFor and , we haveUsing (19) and (ii), for , we obtainThus, we obtainThis is a contradiction. Then, (30) holds. By induction, we can show that, for ,Hence, we can derive for ,By (i) and (37), we obtainwhich implies thatOn the other hand, for any , we haveThen,Letting , we havewhereThis implies that, for all ,Therefore,as desired.
Theorem 2. Assume that , constants , , , , , , , , , , and, and a function satisfying Definition 1 such that(i) and .(ii) and satisfying where and those satisfying(iii) satisfying(iv).(v)For any , .
Then, the solution of system (7) is -stable in -th moment, where and .
Proof. For any , let be the solution of (7) and , where . Since () is satisfied, we can choose sufficiently small such thatLetFor , we obtainBy (iii), we can deriveLet such thatWe will prove that, for ,Using (51), it is obvious to see thatNow, we will prove thatOtherwise, there exists satisfying and , for . By the continuity of in , there exists satisfyingFor and , we obtainThus,Combining (59) with (51) and (ii), for , we obtainThen,which is a contradiction. Hence, (56) holds. Now, we will prove thatIf it is not satisfied, we have . By the continuity of in , there exists satisfyingFor and , we obtainBy (51) and (ii), for and , we haveThen,This is a contradiction. Using (52), (56), and (62), it follows thatMoreover, we can show thatIndeed, there is satisfyingThere exists for . Thus, for and . Then, by (51) and (ii), for , we obtainwhich implies thatwhich is a contradiction. If there is satisfying . By the continuity of in , there exists satisfyingThus, for and ,Then, forwe haveThis is a contradiction. Hence, we can see that . If this is false, we have . To prove this claim, there are two cases to be presented: Case 1: for , . By (53), (56), and (69), we have for and , Thus, by (51) and (ii), for , Then, which implies a contradiction. Case 2: there is satisfying . Since and by the continuity of in , there is satisfyingFor and , we obtainBy (51) and (ii), for , we haveThen,which is a contradiction. By induction, we can show that, for ,Finally, we can derive that, for , .
Proceeding as the proof of Theorem 1, we have and whereTherefore,as desired.
Consider the following NISFDEwMS:wherewhere is arbitrary and . We assume that and such that , is standard one-dimensional Brownian Motion, and is Borel measurable such that , and there exists such thatwhere and
Thus, and .
By the same computation, we have