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# Mean square exponential and non-exponential asymptotic stability of impulsive stochastic Volterra equations

- Dianli Zhao
^{1, 2}Email author and - Dong Han
^{1}

**2011**:9

https://doi.org/10.1186/1029-242X-2011-9

© Zhao and Han; licensee Springer. 2011

**Received:**11 October 2010**Accepted:**17 June 2011**Published:**17 June 2011

## Abstract

In this article, some inequalities on convolution equations are presented firstly. The mean square stability of the zero solution of the impulsive stochastic Volterra equation is studied by using obtained inequalities on Liapunov function, including mean square exponential and non-exponential asymptotic stability. Several sufficient conditions for the mean square stability are presented. Results in this article indicate that not only the impulse intensity but also the time of impulse can influence the stability of the systems. At last, an example is given to show application of some obtained results.

**Mathematics classification** Primary(2000): 60H10, 60F15, 60J70, 34F05.

## Keywords

- Stochastic Volterra equations
- Impulse
- Liapunov function
- Asymptotic stability

## 1 Introduction

Study on the stability of stochastic differential equations has gained lots of attention over the last years. The results and methods have been improved from time to time. Very recently, Taniguchi [1] studied the exponential stability for stochastic delay partial differential equations by use of the energy method which overcomes the difficulty of constructing the Liapunov functional on delay differential equations. Wan and Duan [2] extended the result of Taniguchi [1] to be applied to more general stochastic partial differential equations with memory. Another important method is about the fixed-point theory. It was first used to consider the exponential stability for stochastic partial differential equations with delays by Luo [3], where the conditions do not require the monotone decreasing behavior of the delays. This method also employed in Sakthivel and Luo [4, 5] to study the asymptotic stability of the nonlinear impulsive stochastic differential equations and the impulsive stochastic partial differential equations with infinite delays.

*k*(

*t*) is continuous, integrable and of a single sign. Brauer [6] showed that the solution could not be stable if , Burton and Mahfoud [7] proved the zero solution is asymptotically stable if , Kordonis and Philos [8] discussed the stability of the solution under condition . Therefore, a necessary condition for for all solutions is that . About exponential asymptotic stability, Murakami [9] showed that the uniform asymptotic stability and the exponential asymptotic stability of the zero solution of this equation are equivalent if and only if for some

*γ*> 0. Hence if it fails to hold, a uniformly asymptotically stable solution cannot be exponentially asymptotically stable. Some deeper related work on deterministic equations by Appleby can be found in [10–12], including the so-called "non-exponential decay rate" and "subexponential solution". Mao [13] investigated the mean square stability of the generalized equation

for all *i* ∈ *N* = {0, 1, 2, ·····} by using Liapunov function, which show that both the presence of impulses and the time of the presence can influence the stability of the systems. By choosing the impulse intensity and the impulse time, We find that
is not necessary condition for the exponential asymptotic stability.

The article is organized as follows: some preliminary notations and useful lemmas are given in Sect. 2. Then, sufficient conditions of the mean square exponential asymptotic stability are shown in the first part of Sect. 3, and the second part mainly deals with the mean square non-exponential asymptotic stability of the solution. Finally, an example is given.

## 2 Preliminary notes

*τ*

_{ i },

*i*= 1, 2,...} be a series of numbers such that

*t*

_{0}=

*τ*

_{0}< ···<

*τ*

_{k}<

*τ*

_{k+1}< ··· and . We denote R

^{+}= [0, +1). Consider the impulsive stochastic Volterra equations

where *D*_{
i
} = (*τ*_{I}, *τ*_{i+1}) for all *i* ∈ *N*. *f*(*t*, *x*, *y*) : *R*^{+} × *R*^{
n
} × *R*^{
n
} → *R*^{
n
}, *g*(*t*, *x*, *y*) : *R*^{+} × *R*^{
n
} × *R*^{
n
} → *R*^{
n
}. *ξ*_{
i
} = *τ*_{
i
} - *τ*_{i-1},
with respect to probability distribution for all *i* = 1, 2,.... *I*_{
i
}(*t*, *x*) : *R*^{+} × *R*^{
n
} → *R*^{
n
}. *F* (*t*) and *G*(*t*) are both continuous and integrable matrix-valued functions on R^{+}. B(t) is standard *n*-dimensional Brownian motion on a complete filtered probability space Ω, *F*, (*F*^{
B
} (*t*)) _{t ≥ 0}, *P*), where the filtration is defined as *F* ^{
B
}(*t*) = *σ* (*B*(*s*) : 0 ≤ *s* ≤ *t*). Almost sure events are Palmost sure in this article denoted by "a.s.". Suppose *f*(*t*, 0, *y*) = 0, *g*(*t*, 0, *y*) = 0 and *I*_{
i
}(*t*, 0) = 0 for *t* > *t*_{0}, then *x*(*t*) ≡ 0 is the solution of (1), which is called zero solution of (1). In this article, we always assume there exists a unique stochastic process satisfying (1), and assume all solutions of (1) are continuous on the left and limitable on the right. We further recall the various standard notions of stability of the zero solution required.

**Definition 2.1**.

*The zero solution of*(1)

*is said to be*

- (i)
*mean square asymptotically stable, if for any ε*> 0,*there exist constants δ*> 0*and T*=*T*(*t*_{0},*ε*) > 0*such that E*(||*x*(*t*)||^{2}) <*ε for all t*>*t*_{0}+*T when E*(||*x*_{0}||^{2}) <*δ*. - (ii)
*mean square exponentially asymptotically stable, if for any t*_{0}∈*R*^{+}*there exist λ*> 0,*T*> 0*and C*=*C*(*x*_{0},*t*_{0}) > 0*such that E*(||*x*(*t*)||^{2})<*C*exp (-*λt*)*for t*>*T*. - (iii)
*mean square non-exponentially asymptotically stable*,*if**and**hold*.

*a*∨

*b*=

*max*{

*a*,

*b*},

*E*(

*x*) is the expectation of

*x*and ||

*x*(

*t*)|| is some norm in the sequel. Let

*C*

^{1}[0, ∞) be the family of all continuous functions on [0, ∞) which are once continuously differentiable and

*C*

^{1,2}(

*R*

^{+}×

*R*

^{n},

*R*

^{+}) denote the family of all nonnegative functions from

*R*

^{+}×

*R*

^{ n }to

*R*

^{ n }which are once continuously differentiable in

*t*and twice in

*x*. For each

*V*∈

*C*

^{1,2}(

*R*

^{+}×

*R*

^{ n },

*R*

^{+}), we denote

*V*(

*t*) =

*E*(

*V*(

*t*,

*x*(

*t*))),

*V*(

*t*

^{ - }) =

*E*(

*V*(

*t*,

*x*(

*t*

^{-}))) and

where and .

Before going to the main results, let's consider some lemmas about linear Volterra equation without impulses.

*Then z*(*t*) > 0 *and*
*for t* ≥ *s* ≥ 0. *Moreover*,
*implies that z*(*t*) ≤ 1.

*Proof*. Firstly we claim that

*z*(

*t*) > 0 for all

*t*∈ [0, +∞), if not, there exists

*t*> 0 such that . Then we have

*z*(

*t*) > 0 for all . Since , we get that , then there is satisfying

*z*(

*t*) > 0 for all

*t*∈ [0, +∞). Again from (2), we get

By integrating on both sides, we get
for *t* ≥ *s* ≥ 0.

The proof is complete.

**Lemma 2.3**. [[20], Corollary 3.3]

*Under conditions in Lemma 2.2. Let z*(

*t*)

*be solution of (2). Suppose*

*Then z*(*t*) *is nonincreasing on* [0, +∞).

**Lemma 2.4**.

*Suppose k*(

*t*) > 0

*is a function on R*

^{+}.

*a*> 0

*is constant*.

*h*(

*t*) ≥ 0 is a function on

*R*

^{+}.

*Let y*(

*t*)

*satisfy*

*is true for all t* ∈ (*τ*_{
I
}, *τ*_{i+1}).

*z*(

*t*) is solution of (2). Now we prove that

*y*(

*t*) ≤

*p*(

*t*) for all

*t*∈ [

*τ*

_{ i },

*τ*

_{i+1}). If it is not true, then there exist

*t*

_{1}∈ (

*τ*

_{I},

*τ*

_{i+1}) such that

*y*(

*t*

_{1}) >

*p*(

*t*

_{1}). Denote , then and

for all . By combining (6)(8) and (9), we obtain which contradicts with (7). From above all, we arrive at the desired result.

## 3 Main results

In this section, we consider the nonlinear volterra equation with impulsive effect and denote the solution of (1) by *x*(*t*). Several sufficient conditions of mean square stability are presented by comparison method with Liapunov function, which include mean square exponential asymptotic stability and mean square non-exponential asymptotic stability.

### 3.1 Mean square exponential asymptotic stability

**Theorem 3.1**.

*If there exist positive numbers c*

_{1},

*c*

_{2}and

*V*∈

*C*

^{1,2}(

*R*

^{+}×

*R*

^{ n },

*R*

^{+})

*satisfying*

- (i)
*c*_{1}||*x*||^{ p }≤*V*(*t*,*x*) ≤*c*_{2}||*x*||^{ p }; - (ii)

*for any i*= 1, 2,...;

- (iii)
- (iv)
*and*; - (v)
*there exists γ*> 0*such that**and*.

*Then zero solution of (1) is mean square exponentially asymptotically stable*.

*t*∈ [

*τ*

_{i}

_{-1},

*τ*

_{ i }) by mathematical induction for

*i*= 1, 2,.... We stipulate and as

*i*= 1 here and in the sequel. (11) is true for i = 1 immediately from (10). Assume that (11) holds for any

*i*≥ 1, then for

*t*=

*τ*

_{i}we get

Thus by mathematical induction (11) is true for *i* = 1, 2,....

Then the mean square exponential asymptotic stability of (1) inherits from that of solutions of (2) under assumptions (iv) and (v). The proof is complete.

**Corollary 3.2**. *If there exist positive numbers c*_{1}, *c*_{2} and *V* ∈ *C*^{1,2}(*R*^{+} × *R*^{
n
}, *R*^{+}) *satisfying (i)-(iii) and*

*(v) in Theorem 3.1 and*

*(H1)*
exp (*as*)d*s* < ∞*;*

*(H2)* *there exists* 0 < *ρ* < 1 *such that*
.

*Then zero solution of (1) is mean square exponentially asymptotically stable*.

*Proof*. Since
implies
, the result is proved by Theorem 3.1.

**Theorem 3.3**.

*If there exist positive numbers c*

_{1},

*c*

_{2}

*and V*∈

*C*

^{1,2}(

*R*

^{+}×

*R*

^{ n },

*R*

^{+})

*satisfying*

- (i)
*c*_{1}||*x*||^{ p }≤*V*(*t*,*x*) ≤*c*_{2}||*x*||^{ p }; - (ii)

*holds when*

*E*||

*x*(

*τ*

_{i-1})||

^{2}<

*θ*

*for some constant*

*θ*> 0;

- (iii)
- (iv)
*there exists*0 <*ρ*< 1*such that*; - (v)
;

- (vi)
*τ*_{i}≤*t*_{0}+*i for all i*∈*N*.

*Then zero solution of (1) is mean square exponentially asymptotically stable*.

*t*∈ [

*τ*

_{ i-1 }

*, τ*

_{ i }) by mathematical induction. From (12), it is obviously true for

*i*= 1. Assume that (14) is true for any

*i*≥ 1, then for all

*t*∈ [

*τ*

_{ i-1 }

*, τ*

_{ i })), it is true that

for
. Then by mathematical induction (14) is true for *i* = 1,2,....

since *τ*_{i-1}≤ *t* ≤ *τ*_{
i
}≤ *t*_{0} + *i*.

Therefore holds. The proof is complete.

**Remark 1**. *Theorem 3.3 is not a simple corollary of Theorem 3.1, since the conditions (ii) and (v) in Theorem 3.3 is weaker than that in Theorem 3.1*.

**Remark 2**. *Theorem 3.3 shows that*
*is not necessary condition for exponential asymptotical stability, which can also be found in Theorem 3.5*.

### 3.2 Mean square non-exponential asymptotic stability

To show that the solution of (1) is mean square non-exponentially asymptotically stable, we have to prove that and . Now we prove the solution convergent to zero firstly.

**Theorem 3.4**.

*If there exist positive numbers c*

_{1},

*c*

_{2}

*and V*∈

*C*

^{1,2}(

*R*

^{+}×

*R*

^{ n },

*R*

^{+})

*satisfying*

- (i)
*c*_{1}||*x*||^{ p }≤*V*(*t*,*x*) ≤*c*_{2}||x||^{ p }; - (ii)

*holds for some constant a*> 0;

- (iii)
- (iv)
*there exists*0 <*ρ*< 1 such that ; - (v)
.

*Then zero solution of (1) is mean square asymptotically stable*.

*t*∈ [

*τ*

_{i-1},

*τ*

_{ i }). Noticing 0 <

*ρ*< 1, for any

*ε*> 0, there is

*k*

_{0}> 0 such that where . For

*h*(

*t*) is integrable, for any

*ε*defined above, there is such that for . It follows that

for
. By choosing
, it follows (15) directly that for any *ε* > 0, we have
for
when
. The proof is complete.

**Theorem 3.5**. *If there exist positive numbers c*_{1}, *c*_{2} *and V* ∈ *C*^{1,2}(*R*^{+} × *R*^{
n
}, *R*^{+}) *satisfying* (*i*)-(*v*) *in theorem 3.3 and*

*for any* *i* = 1, 2,...;

*(H3)*
*satisfies*
and
;

*(H4) there is constant* 1 > *d* > 0 *such that*
;

*(H5)* log (*τ*_{
i
} - *t*_{0}) ≥ *i for all i* = 1, 2,....

*Then zero solution of (1) is mean square non-exponentially asymptotically stable*.

*t*∈ [

*τ*

_{i-1},

*τ*

_{ i }) for

*i*= 1, 2,... by mathematical induction from (H1)-(H2). From assumption

The proof is complete.

**Remark 3**. *Assumption (H5) in Theorem 3.5 can be replaced by*
.

## 4 Example

*for all k* ∈ *N*. *λ*_{1}(*k*) *and λ*_{2}(*k*) *are random variables on*
. *Then the zero solution of (18) and (19) is mean square non-exponentially asymptotically stable*.

Since
and log *τ*_{
k
} = *k*, we finish the proof by Theorem 3.5.

## Declarations

### Acknowledgements

The authors sincerely thank the anonymous reviewer for his careful reading, constructive comments and fruitful suggestions to improve the quality of the manuscript. This article is partially supported by NSFC (No. 11001173).

## Authors’ Affiliations

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