 Research
 Open Access
Stability of sets of stochastic functional differential equations with impulse effect
 Yan Xu^{1} and
 Zhimin He^{1}Email author
https://doi.org/10.1186/s1366201505512
© Xu and He 2015
 Received: 5 December 2014
 Accepted: 22 June 2015
 Published: 4 July 2015
Abstract
In this paper, we study the stability of sets for a class of impulsive stochastic functional differential equations. By employing piecewise continuous Lyapunov functions with Razumikhin methods, some sufficient conditions are established to guarantee the stability of sets of impulsive stochastic functional differential equations and we also show that the impulses play an important role in the stability of stochastic functional differential equations. Three examples are presented to illustrate the effectiveness of the results obtained.
Keywords
 stability of sets
 Brownian motion
 stochastic functional differential equations
 impulse
 Lyapunov function
 Razumikhin methods
MSC
 34K20
 34K45
 34K50
1 Introduction
During the past few decades, the stability theory of stochastic differential equations and impulsive differential equations has been developed very quickly; see for instance [1–15]. A lot of stability criteria on impulsive stochastic differential equations have also been reported (see [16–23] and the references therein). Almost all of them mainly focus on the stability of the zero solution, but there is very little of research addressing the stability of sets.
The concept of stability of sets of nonlinear systems, which includes as a special case stability in the sense of Lyapunov (see Krasovskii [24]; Rouche et al. [25]), such as stability of the trivial solution, stability of the solution, stability with respect to part of the variables and so on, has become one of the most important issues in the stability theory of nonlinear systems [26–28]. The theoretical works of the stability of sets with respect to nonlinear ordinary differential equations may be traced back to Yoshizawa [29–31] in the previous century. The research to the stability of sets of impulsive differential equations can be found in [15, 32–35]. For stochastic differential equations and impulsive stochastic differential equations, we refer the reader to [11, 36–39] and the references therein.
In this paper, we shall extend the Razumikhin method developed in [7, 14, 40] to investigate the stability of sets for a class of impulsive stochastic functional differential equations. Meanwhile, our results show that the impulsive effects play an important part in the stability for stochastic functional differential equations, that is, an unstable stochastic delay system can be successfully stabilized by impulses.
The rest of this paper is organized as follows. Some preliminary notes are given in Section 2. Several theorems on stability of sets of impulsive stochastic functional differential equation are established in Section 3. In Section 4, three examples are presented to illustrate the applications of the results obtained.
2 Preliminaries
Throughout this paper, we use the following notations.
Let \((\Omega, \mathcal{F}, \{\mathcal{F}_{t}\}_{t\geq0}, P)\) be a complete probability space with a natural filtration \(\{\mathcal {F}_{t}\}_{t\geq0}\) satisfying the usual conditions (i.e. it is right continuous and \(\mathcal{F}_{0}\) contains all Pnull sets), and \(E[\cdot]\) stand for the correspondent expectation operator with respect to the given probability measure P. Let \(W(t)=(W_{1}(t), \ldots, W_{m}(t))^{T}\) be an mdimensional Wiener process defined on a complete probability space with a natural filtration. Let \(\cdot\) denote the Euclidean norm in \(\mathbb{R}^{n}\).
Let \(\tau>0\) and \(PC([\tau,0]; \mathbb{R}^{n})\) = {\(\phi:[\tau ,0]\rightarrow\mathbb{R}^{n}\mid\phi(t)\) is continuous everywhere except at the points \(t=t_{k}\in[t_{0},\infty)\), \(\phi(t_{k}^{+})\) and \(\phi(t_{k}^{})\) exist with \(\phi(t_{k}^{+})=\phi(t_{k})\)} with the norm \(\\phi\=\sup_{\tau\leq\theta\leq0}\phi(\theta)\), where \(\phi(t^{+})\) and \(\phi(t^{})\) denote the righthand and lefthand limits of function \(\phi(t)\) at t.
Denote \(PC_{\mathcal{F}_{0}}^{b}([\tau,0]; \mathbb{R}^{n})\) by the family of all bounded, \(\mathcal{F}_{0}\)measurable, \(PC([\tau,0]; \mathbb{R}^{n})\)valued random variables. For \(p>0\), denote by \(PC_{\mathcal{F}_{t}}^{p}([\tau,0]; \mathbb{R}^{n})\) the family of all \(\mathcal{F}_{t}\)measurable \(PC([\tau,0]; \mathbb{R}^{n})\)valued random variables ϕ such that \(E\\phi\^{p}<\infty\).
Definition 2.1
 (i)
x: \([\sigma\tau, \sigma+\beta)\) for some β (\(0<\beta\leq \infty\)) is continuous for \(t\in[\sigma\tau, \sigma+\beta)\backslash\{ t_{k}: k=1,2,\ldots\}\), \(x(t^{+}_{k})\) and \(x(t^{}_{k})\) exist with \(x(t^{+}_{k})=x(t_{k})\) for \(t_{k}\in[\sigma\tau, \sigma+\beta)\), and \(\{x_{t}\}_{t\geq t_{0}}\) is \(\mathcal{F}_{t}\)adapted;
 (ii)
\(\{f(t,x_{t})\}\in L^{1}([t_{0},\infty];\mathbb{R}^{n})\) and \(\{ g(t,x_{t})\}\in L^{2}([t_{0},\infty];\mathbb{R}^{n\times m})\);
 (iii)
\(x(t)\) satisfies (2.1).
We denote the solution of the initial problem (2.1) by \(x(t;\sigma,\xi)\), and we denote by \([\sigma\tau, \sigma+\beta)\) the maximal right interval in which the solution \(x(t;\sigma,\xi)\) is defined.
 (H_{1}):

For all \(\psi\in PC([\tau,0]; \mathbb{R}^{n})\) and \(k\in\mathbb{Z}^{+}\), the limitsexist.$$\lim_{(t,\varphi)\rightarrow(t^{}_{k},\psi)}f(t,\varphi )=f\bigl(t^{}_{k}, \psi\bigr), \qquad \lim_{(t,\varphi)\rightarrow(t^{}_{k},\psi )}g(t,\varphi)=g\bigl(t^{}_{k}, \psi\bigr) $$
 (H_{2}):

f and g satisfy the locally Lipschitz condition in ϕ on each compact set in \(PC([\tau,0]; \mathbb{R}^{n})\). More precisely, for every \(a\in[t_{0}, \sigma+\beta)\) and every compact set \(G\in PC([\tau,0]; \mathbb{R}^{n})\), there exists a constant \(L=L(a,G)\) such thatwhenever \(t\in[t_{0}, a)\) and \(\varphi, \psi\in G\).$$\bigl\vert f(t,\varphi)f(t,\psi)\bigr\vert \vee\bigl\vert g(t, \varphi)g(t,\psi)\bigr\vert \leq L\\varphi \psi\, $$
 (H_{3}):

For any \(\rho>0\) there exists \(0<\rho_{1}\leq\rho\), such thatfor all \(k\in\mathbb{Z}^{+}\).$$x\in M(t,\rho_{1}) \quad \mbox{implies that} \quad x+I_{k}(t_{k},x)\in M(t,\rho) $$
 (H_{4}):

\(f(t,x_{t}), g(t,x_{t})\in PC([t_{0}, \infty), \mathbb {R}^{n})\) for \(x_{t}\in PC([\sigma\tau, \infty), \mathbb{R}^{n})\).
For any \(t\geq t_{0}\) and \(\kappa\geq0\), let \(PC_{\kappa}=\{\phi\in PC([\tau,0]; \mathbb{R}^{n}): \\phi\\leq\kappa\}\).
 (A_{1}):

for each \(t\in[t_{0}, \infty)\) the set \(M(t)\) is not empty;
 (A_{2}):

for any compact subset F of \([t_{0}, \infty)\times \mathbb{R}^{n}\) there exists a constant \(K>0\) depending on F such that if \((t,x), (t',x)\in F\), then the following inequality holds:$$\bigl\vert d\bigl(x,M(t)\bigr)d\bigl(x,M\bigl(t'\bigr)\bigr) \bigr\vert \leq K\bigl\vert tt'\bigr\vert ; $$
 (A_{3}):

if for solution \(x(t;\sigma,\xi)\) there exists \(h>0\) satisfyingwhere ρ is a constant, then \(x(t;\sigma,\xi)\) is defined in the interval \([\sigma,\infty)\).$$d\bigl(x(t;\sigma,\xi),M(t,\rho)\bigr)\leq h< \infty\quad \mbox{for } t\in [ \sigma,\sigma+\beta), $$
Definition 2.2
 (B_{1}):

V is continuous on each of the set \(([t_{0}\tau ,t_{0}]\cup[t_{k1},t_{k}))\times M(t,\rho)\) for all \(x\in M(t,\rho)\) and for \(k\in\mathbb{Z}^{+}\), the limit \(\lim_{(t,y)\rightarrow (t^{}_{k},x)}V(t,y)=V(t^{}_{k},x)\) exists;
 (B_{2}):

V is locally Lipschitz in \(x\in M(t,\rho)\), \(V(t,0)=0\) for \((t,x)\in M\) and \(V(t,x)>0\) for \((t,x)\notin M\).
Definition 2.3
We shall give the definitions of stability of the set M with respect to system (2.1).
Definition 2.4
 (S_{1}):

stable, if for any \(\sigma\geq t_{0}\), \(\alpha>0\), and \(\epsilon>0\), there is a \(\delta(\sigma,\epsilon,\alpha)>0\) such that \(\xi\in PC_{\alpha}\cap M_{0}(\sigma,\delta)\) implies that \(x(t,\sigma ,\xi)\in M(t,\epsilon)\) for \(t\geq\sigma\);
 (S_{2}):

uniformly stable, if the δ in (S_{1}) is independent of σ;
 (S_{3}):

asymptotically stable, if it is stable and for any \(\sigma \geq t_{0}\) and \(\alpha>0\), there exists a \(\delta=\delta(\sigma,\alpha )\) such that \(\xi\in PC_{\alpha}\cap M_{0}(\sigma,\delta)\) implies that \(x(t,\sigma,\xi)\rightarrow M(t)\) as \(t\rightarrow\infty\);
 (S_{4}):

uniformly asymptotically stable, if it is uniformly stable, and for any \(\alpha>0\) there exists a \(\delta(\alpha)>0\), such that for any \(\epsilon>0\) there is a \(T(\epsilon,\alpha,\delta)>0\) such that \(\sigma\geq t_{0}\) and \(\xi\in PC_{\alpha}\cap M_{0}(\sigma,\delta )\) implies that \(x(t,\sigma,\xi)\in M(t,\epsilon)\) for \(t\geq\sigma+T\).
3 Main results
In this section, we present and prove our main results on uniform stability and asymptotic stability of the sets of system (2.1) by utilizing piecewise continuous Lyapunov functions with Razumickhin methods.
Theorem 3.1
 (i)
\(a(d(x,M(t)))\leq EV(t,x)\leq b(d(x,M(t)))\) for all \((t,x)\in[t_{0}\tau, \infty)\times M(t,\rho)\);
 (ii)
\(ELV(t,x(t))\leq\eta(t)c(EV(t,x(t)))\), \(t\neq t_{k}\), whenever \(EV(t+s,x(t+s))\leq P(EV(t,x(t)))\) for \(\tau\leq s\leq0\), where \(x(t)\) is any solution of system (2.1), and \(\eta:[t_{0},\infty)\rightarrow\mathbb{R}^{+}\) is locally integrable;
 (iii)
\(EV(t_{k},x+I_{k}(t_{k},x))\leq P^{1}(EV(t^{}_{k},x))\) for each \(k\in\mathbb{Z}^{+}\), and all \(x\in M(t, \rho_{1})\), where \(P^{1}\) is the inverse of the function P;
 (iv)
\(\sup_{k\in\mathbb{Z}^{+}}\{t_{k}t_{k1}\}<\infty \), and \(\int_{P^{1}(\mu)}^{\mu}\frac{ds}{c(s)}\int_{t_{k1}}^{t_{k}}\eta (s)\, ds>0\) for all \(\mu\in(0,\infty)\), \(k\in\mathbb{Z}^{+}\).
Then the set M is uniformly stable with respect to the solution of system (2.1).
Proof
For any given \(\epsilon>0\), \(\alpha>0\), without loss of generality, we assume that \(\epsilon\leq\rho_{1}\). We can choose \(\delta=\delta(\epsilon,\alpha)>0\) such that \(P(b(\delta ))<\alpha(\epsilon)\) and \(\delta<\alpha\). From \(b(\delta)< P(b(\delta))<\alpha(\epsilon)<b(\epsilon)\) we know that \(\delta<\epsilon\).
Next, we will prove \(d(x(t),M(t))<\epsilon\) for \(t\in[\sigma,\sigma +\beta)\). Suppose, on the contrary, that \(d(x(t),M(t))>\epsilon\) for some \(t\in [\sigma,\sigma+\beta)\). Then let \(\hat{t}=\inf\{\sigma\leq t\leq\sigma+\beta \mid d(x(t),M(t))>\epsilon\}\). Note that \(d(x(\sigma),M(\sigma))<\epsilon\), we see that \(\hat{t}>\sigma\), \(d(x(t),M(t))\leq\epsilon\leq\rho_{1}\), for \(t\in[\sigma\tau,\hat{t})\) and either \(d(x(\hat{t}),M(\hat {t}))=\epsilon\) or \(d(x(\hat{t}),M(\hat{t}))>\epsilon\) and \(\hat {t}=t_{k}\) for some k.
Now let us first consider the case \(t_{m1}\leq\tilde{t}< t_{m}\). Let \(\bar{t}=\sup\{t\in[\sigma,\tilde{t}] \mid EV(t)\leq P^{1}(a(\epsilon))\}\). Since \(EV(\sigma)< P^{1}(a(\epsilon))\), \(EV(\tilde {t})=a(\epsilon)>P^{1}(a(\epsilon))\), and \(EV(t)\) is continuous on \([\sigma,\tilde{t}]\), we have \(\bar{t}\in(\sigma,\tilde{t})\), \(EV(\bar {t})=P^{1}(a(\epsilon))\), and \(EV(t)\geq P^{1}(a(\epsilon))\) for \(t\in [\bar{t},\tilde{t}]\).
Remark 3.1
From Theorem 3.1, we know that impulsive perturbations may cause uniform stability even if the unperturbed system is unstable.
The following result on the asymptotical stability of sets will reveal that impulsive perturbation make stable systems asymptotically stable.
Theorem 3.2
 (i)
\(a(d(x,M(t)))\leq EV(t,x)\leq b(d(x,M(t)))\) for all \((t,x)\in[t_{0}\tau, \infty)\times M(t,\rho)\);
 (ii)
\(EV(t_{k},x+I_{k}(t_{k},x))EV(t^{}_{k},x)\leq h_{k}(EV(t^{}_{k},x))\) for all \(k\in\mathbb{Z}^{+}\) and \(x\in M(t,\rho_{1})\);
 (iii)
for any solution \(x(t)\) of system (2.1), \(ELV(t,x)\leq0\);, and for any \(\sigma\geq t_{0}\), and \(r>0\), there exists \(\{r_{k}\}\) such that \(EV(t,x)\geq r\) for \(t\geq \sigma\) implies that \(h_{k}(EV(t^{}_{k},x))\geq r_{k}\); where \(r_{k}\geq0\) with \(\sum_{k=1}^{\infty}r_{k}=\infty\).
Then the set M with respect to the solution of system (2.1) is uniformly stable and asymptotically stable.
Proof
At first, we show that the set M is uniform stability.
For given \(\epsilon>0 \) (\(\epsilon\leq\rho_{1}\)), \(\alpha>0\), we choose a \(\delta(\epsilon,\alpha)>0\) such that \(b(\delta)\leq a(\epsilon )\) and \(\delta<\alpha\). For any \(\sigma\geq t_{0}\) and \(\xi\in PC_{\alpha}\cap M_{0}(\sigma ,\delta)\), let \(x(t)=x(t;\sigma,\xi)\) be the solution of system (2.1). We will show that \(x(t)\in M(t,\epsilon)\) for \(t\in[\sigma ,\sigma+\beta)\).
Set \(EV(t)=EV(t,x(t))\), where \(\sigma\in[t_{m1},t_{m})\) for some \(m\in\mathbb{Z}^{+}\). Then condition (iii) implies that \(ELV(t)\leq0\) for \(t\in[\sigma,\sigma +\beta)\cap([\sigma,t_{m})\cup(\bigcup_{k=m}^{\infty}[t_{k1},t_{k})))\), \(k\in\mathbb{Z}^{+}\).
Next we shall prove that the set M is asymptotically stable.
From conditions (ii), (iii), and \(EV(t)\geq0\), we note that \(EV(t)\) is nonincreasing on the interval \([\sigma,\infty)\). So the limit \(\lim_{t\rightarrow\infty}EV(t)\) exists.
Assume \(\sigma\in[t_{m1},t_{m}]\) for some \(m\in\mathbb{Z}^{+}\). Set \(\lim_{t\rightarrow\infty}EV(t)=r\geq0\), one can easily see that \(EV(t)\geq r\) for \(t\geq\sigma\). Then by condition (iii), it follows that there is a sequence \(\{r_{k}\}\) with \(r_{k}\geq0\) for \(k\in\mathbb {Z}^{+}\), which implies that \(h_{k}(EV(t^{}_{k},x))\geq r_{k}\) with \(\sum_{k=1}^{\infty}r_{k}=\infty\).
Theorem 3.3
 (i)
\(a(d(x,M(t)))\leq EV(t,x)\leq b(d(x,M(t)))\) for all \((t,x)\in[t_{0}\tau, \infty)\times M(t,\rho)\);
 (ii)
\(EV(t_{k},x+I_{k}(t_{k},x))\leq\psi _{k}(EV(t^{}_{k},x))\), for all \(K\in\mathbb{Z}^{+}\), and \(x\in M(t,\rho)\);
 (iii)for any solution \(x(t)\) of system (2.1), \(ELV(t,x)\leq\theta(t)C(EV(t,x))\) for \(t\neq t_{k}\), where \(\theta:[t_{0},\infty)\rightarrow\mathbb{R}^{+}\) is locally intergrade, and there exists \(\mu_{0}\), such that for any \(\mu\in(0,\mu _{0})\),where \(\gamma_{k}\geq0\) with \(\sum_{k=1}^{\infty}\gamma_{k}=\infty\).$$\int_{\mu}^{\psi_{k}(\mu)}\frac{ds}{C(s)}\int _{t_{k1}}^{t_{k}}\theta (s)\, ds\leq\gamma_{k}, $$
Then the set M with respect to the solution of system (2.1) is uniformly stable and asymptotically stable.
Proof
Without loss of generality, for any given \(\epsilon >0\), \(\alpha>0\), we can assume that \(\epsilon\leq\rho_{1}\). We choose a \(\beta:0<\beta<\min\{a(\epsilon),\mu_{0}\}\) such that \(\psi _{k}(s)< a(\epsilon)\) for \(0\leq s\leq\beta\) and for all \(k\in\mathbb{Z}^{+}\).
Since \(EV(t)\leq EV(t_{k})\) for \(t_{k}\leq t< t_{k+1}\), it follows that \(\lim_{t\rightarrow\infty}EV(t)=0\), which yields \(\lim_{t\rightarrow \infty}d(x(t,M(t)))=0\). The proof of Theorem 3.3 is complete. □
4 Illustrative examples
As an application, we consider the following examples.
Example 4.1

\(t_{k}t_{k1}<\frac{\ln0.5}{1.6}\), for \(k\in \mathbb{Z}^{+}\), where \(t_{0}\geq0\).
Example 4.2

\(0< u<1\) and \(m+nu^{1}+\frac {1}{2}p^{2}+pqu^{1}+\frac{1}{2}q^{2}u^{2}<0\).
We check that for any \(\sigma\geq t_{0}\), and \(r>0\), there exists \(\{r_{k}\}\) such that \(EV(t,x)\geq r\) for \(t\geq\sigma\) implies that \(h_{k}(EV(t^{}_{k},x))\geq r_{k}\); where \(r_{k}\geq0\) with \(\sum_{k=1}^{\infty}r_{k}=\infty\). Since \(h_{k}(EV(t^{}_{k},x))=(1u^{2})EV(t^{}_{k},x)\), when \(EV(t,x)\geq r\) for \(t\geq\sigma\), then we have \(h_{k}(EV(t^{}_{k},x))=(1u^{2})EV(t^{}_{k},x)\geq(1u^{2})r\). We take \(r_{k}=(1u^{2})r\) and we have \(\sum_{k=1}^{\infty}r_{k}=\infty\). Thus all of the conditions in Theorem 3.2 are satisfied. Therefore, it follows from Theorem 3.2 that the set M is uniformly stable and asymptotically stable with respect to the solution of the system (4.2).
Example 4.3
 (i)
\(0< h<1\) and \(a+bh^{1}+\frac {1}{2}c^{2}+crh^{1}+\frac{1}{2}r^{2}h^{2}<0\);
 (ii)
\(t_{k}t_{k1}>\frac{\ln h}{a+bh^{1}+\frac {1}{2}c^{2}+crh^{1}+\frac{1}{2}r^{2}h^{2}}\), for \(k\in\mathbb {Z}^{+}\), where \(t_{0}\geq0\).
Declarations
Acknowledgements
The authors would like to thank the referee for his/her careful reading and valuable suggestions, which lead to improvement of the manuscript.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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