Exponential stability of impulsive stochastic partial differential equations with delays

Accepted Manuscript Exponential stability of impulsive stochastic partial differential equations with delays Dingshi Li, Xiaoming Fan

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S0167-7152(17)30116-5 http://dx.doi.org/10.1016/j.spl.2017.03.016 STAPRO 7894

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Statistics and Probability Letters

Received date : 1 April 2015 Revised date : 8 March 2017 Accepted date : 14 March 2017 Please cite this article as: Li, D., Fan, X., Exponential stability of impulsive stochastic partial differential equations with delays. Statistics and Probability Letters (2017), http://dx.doi.org/10.1016/j.spl.2017.03.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Exponential stability of impulsive stochastic partial differential equations with delays∗ Dingshi Li,† Xiaoming Fan b

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, P.R. China

Abstract In this paper, we are concerned with the exponential stability problem of impulsive control stochastic partial differential equations with delays. By employing the formula for the variation of parameters and inequality technique, several criteria on exponential stability are derived and the exponential convergence rate is estimated. Some examples are given to illustrate the theoretical results and to show that the criteria can be applied to stabilize the continuous system with delays, which may be unstable. Keywords: Impulsive control; Delays; Stochastic; Exponential stability

1

Introduction Recently, there has been increasing interest in the study of the existence, uniqueness and stability of mild

solutions of stochastic partial functional differential equations due to their range of applications in various sciences such as physics, mechanical, engineering, control theory and economics, and many significant results have been obtained, see, for example, [1–10]. On the other hand, impulsive control systems arise naturally in a wide variety of applications involving impulsive control for dosage supply in pharmacokinetics, ecosystems management, stabilization and synchronization in chaotic secure communication systems and other chaos systems. Therefore, an interesting subject is to discuss the stability of mild solutions of stochastic partial functional differential equations. In recent years, although a large number of stability criteria of these systems have been reported [11–13], all stability criteria requires the stability of the corresponding continuous stochastic system. In this paper, we are concerned with the exponential stability problem of impulsive control stochastic partial differential equations with delays. By employing the formula for the variation of parameters and inequality technique, several criteria on exponential stability are derived and the exponential convergence ∗ The

work is supported by National Natural Science Foundation of China under Grant 11271270 and 11601446, Chinese

Visiting Scholar abroad program, Fundamental Research Funds for the Central Universities under Grant 2682015CX059. † Corresponding authors: [email protected](D. Li).

1

rate is estimated. The exponential stability criteria do not require the stability of the corresponding continuous stochastic system, and so can be more widely applied to stabilize the continuous stochastic system with delays, which may be unstable, by using impulsive control. Two examples are given to illustrate the effectiveness of the results.

2

Model description and preliminaries Throughout this paper, H and K will denote two real separable Hilbert spaces and we denote by h·, ·iH ,

h·, ·iK their inner products and by k·kH , k·kK their vector norms, respectively. Let (Ω, F , {Ft }t≥0 , P ) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (i.e, it is right continuous and F0 contains all P-null sets). We denote by L(K, H) the set of all linear bounded operators from K into H, equipped with the usual operator norm k·k . In this paper, we always use the same symbol k·k to denote norms of operators regardless of the spaces potentially involved when no confusion possibly arises. E[f ] means the mathematical expectation of f Let {W (t), t ≥ 0} denote a K-valued {Ft }t≥0 -Wiener process defined on (Ω, F , {Ft }t≥0 , P ) with covariance operator Q, i.e., EhW (t) , xiK hW (s) , yiK = (t ∧ s) hQx, yiK

for all x, y ∈ K,

where Q is a positive, self-adjoint, trace class operator on K. In particular, we shall call such W (t), t ≥ 0, a K-valued Q-Wiener process with respect to {Ft }t≥0 . In order to define stochastic integrals with respect to the Q-Wiener process W (t), we introduce the

 subspace K0 = Q1/2 (K) of K which, endowed with the inner product hu, viK0 = Q−1/2 u, Q−1/2 v K , is a Hilbert space. Let L02 = L2 (K0 , H) denote the space of all Hilbert-Schmidt operators from K0 into H. It turns out to be a separable Hilbert space, equipped with the norm 2

kψkL0 = tr 2

  ∗  ψQ1/2 ψQ1/2

for all ψ ∈ L02 .

Clearly, for any bounded operators ψ ∈ L(K, H), this norm reduces to kψk2L0 = tr (ψQψ ∗ ). The reader is 2

referred to Da Prato and Zabczyk [14] for a systematic theory about stochastic integrals of this kind. R+ = [0, +∞), N = {1, 2, 3, . . .} and C(X, Y ) denotes the space of continuous mappings from the topological space X to the topological space Y . Let γ(t), δ(t) ∈ C(R+ , [0, τ ]), where τ ≥ 0. n P C(J, F ) = ψ(t) : J → F | ψ(t) is continuous for all but tk ∈ R

o − + and at these points tk ∈ R, ψ(t+ k ) and ψ(tk ) exist, ψ(tk ) = ψ(tk ) ,

where J ⊂ R is an interval, F is a complete metric space, ψ(s+ ) and ψ(s− ) denote the right-hand and . left-hand limit of the function ψ(s), respectively, the fixed moments of time tk , k = 1, 2, . . ., satisfies 0 = ∆

t0 < t1 < t2 < . . . < tk < . . . , and lim tk = ∞. Especially, let P C = P C ([−τ, 0] , H) equipped with the k→∞

supremum norm kϕkP C =

sup kϕ (s)kH . For φ ∈ P C([−τ, 0] , R), we denote |φ(t)|τ =

−τ ≤s≤0

2

sup |φ(t + s)|.

−τ ≤s≤0

Denote by P CF0 ([−τ, 0] , H) the family of all bounded F0 -measurable, P C-valued random variables φ, p

satisfying kφkLp =

p

sup Ekφ (s)kH < ∞ , where p ≥ 2.

−τ ≤s≤0

Consider impulsive stochastic partial differential equations with delays:    dx (t) = (Ax (t) + g (t, x(t − γ(t)))) dt + σ (t, x(t − δ(t))) dW (t) ,   ∆x (tk ) = ck x(t− t = tk , k ∈ N, k ),     x (s) = ϕ ∈ P C b ([−τ, 0] , H) , s ∈ [−τ, 0] , 0

6 t > 0, t = tk , (1)

F0


where f, g : [0, ∞) × H → H and σ : [0, ∞) × H → L02 are jointly continuous functions, ∆x (tk ) = x t+ k −
x t− k denotes the jump in the state x at time tk , tk is fixed, ck are constants, and A : D (A) ⊂ H → H is the

infinitesimal generator of a semigroup of bounded linear operator (T (t))t≥0 on a Hilbert space H satisfying kT (t)k ≤ M eγt ,

t ≥ 0,

for some constants M ≥ 1 and γ. We always assume that the following condition is satisfied. (H) There exist constants Lg > 0 and Lσ > 0 such that for any x, y ∈ H and t ≥ 0, kg (t, x) − g (t, y)kH ≤ Lg kx − ykH , kσ (t, x) − σ (t, y)kL0 ≤ Lσ kx − ykH , 2

kg (t, 0)kH = 0, kσ (t, 0)kL0 = 0. 2

Then Eq. (2.1) obviously has a trivial solution when ϕ ≡ 0. Definition 2.1. A stochastic process x(t), t ≥ 0 is a mild solution of (1), if (i) x(t) is Ft -adapted, t ≥ 0; (ii) x(t) has c` adl` ag paths on t ≥ 0 almost surely and satisfies the integral equation Z t Z t x (t) = T (t) x (0) + T (t − s)g (s, x(s − γ(s))) ds + T (t − s)σ (s, x(s − δ(s))) dW (s) 0 0 X + T (t − tk )ck x(t− k ), t ≥ 0, tk
b where x0 (s) = ϕ ∈ P CF ([−τ, 0] , H) . 0

Lemma 2.1. [16, Theorem 5.3] Assume that (H) holds. Then the initial value problem (1) has a unique mild solution x(t), t ≥ 0. Remark 2.1. Since the condition (H) is global Lipschitz condition, the condition (5.4) and (5.5) in [16] is not needed. Lemma 2.2. Assume that (H) holds. The mild solution of (1) can be represented by Z t x (t) = K (t, 0) x (0) + K (t, s)g (s, x(s − γ(s))) ds +

Z

0

t

0

where K (t, s) =

   T (t − s),  

K (t, s)σ (s, x(s − δ(s))) dW (s) ,

t ≥ 0,

t, s ∈ [tk−1 , tk ) ,

(1 + ck ) T (t − s), tk−1 ≤ s < tk ≤ t < tk+1 ,     Qk (1 + c ) T (t − s), t j i−1 ≤ s < ti < tk ≤ t < tk+1 . j=i 3

Proof. By the definition 2.1, we have x (t) =

Z

t

T (t) x (0) + T (t − s)g (s, x (s − γ (s))) ds 0 Z t + T (t − s)σ (s, x (s − δ (s))) dW (s) 0

=

Z t K (t, 0) x (0) + K (t, s)g (s, x (s − γ (s))) ds 0 Z t + K (t, s)σ (s, x (s − δ (s))) dW (s) , t ∈ [0, t1 ) . 0

Suppose that for all m = 1, 2, . . . , k Z t x (t) = K (t, 0) x (0) + K (t, s)g (s, x (s − γ (s))) ds 0 Z t + K (t, s)σ (s, x (s − δ (s))) dW (s) , t ∈ [tm−1 , tm ) . 0

Then we obtain that for m = k + 1,  Z (1 + ck ) K (tk , 0) x (0) +

tk

K (tk , s)g (s, x (s − γ (s))) ds  Z tk + K (tk , s)σ (s, x (s − δ (s))) dW (s) T (t − tk )

x (t) =

0

+ +

Z

0 t

tk Z t tk

=

T (t − s)g (s, x (s − γ (s))) ds T (t − s)σ (s, x (s − δ (s))) dW (s)

K (t, 0) x (0) + +

Z

0

Z

t

0

t

K (t, s)g (s, x (s − γ (s))) ds

K (t, s)σ (s, x (s − δ (s))) dW (s) ,

t ∈ [tk , tk+1 ) .

By the mathematical induction, the proof is complete. Lemma 2.3. Given a constant η ≥ k1 + ck k for all k ∈ N , we have the following: (1) if 0 < η < 1 and ρ = sup {tk − tk−1 } < ∞, then k∈N

ln η 1 kK (t, s)k ≤ M e(γ+ ρ )(t−s) , η

0 ≤ s ≤ t;

(2) if η ≥ 1 and θ = inf {tk − tk−1 } > 0, then k∈N

kK (t, s)k ≤ M ηe(γ+

ln η θ

)(t−s) ,

0 ≤ s ≤ t.

Proof. By the induction, we easily obtain 

kK (t, s)k ≤ M 

Y

s


|1 + ck | eγ(t−s) ,

0 ≤ s ≤ t.

Case (1): we have from 0 < η < 1 and ρ = sup {tk − tk−1 } < ∞ that k∈N

kK (t, s)k ≤ M η

t−s ρ −1

ln η 1 eγ(t−s) ≤ M e(γ+ ρ )(t−s) . η

4

Case (2): we have from η ≥ 1 and θ = inf {tk − tk−1 } > 0 that k∈N

kK (t, s)k ≤ M η

t−s θ +1

eγ(t−s) ≤ M ηe(γ+

ln η θ

)(t−s) .

The proof is complete. b Later on we shall often denote the solution of (1) by x(t) = x(t, 0, ϕ) for all ϕ ∈ P CF ([−τ, 0] , H). 0

Definition 2.2. The zero solution of system (1) is said to be p-exponentially (p ≥ 2) stable if there exist

b positive constants λ > 0 and M1 > 1, for any initial value ϕ ∈ P CF ([−τ, 0] , H) such that 0 p

p

Ekx (t)kH ≤ M1 kϕkLp e−λt ,

t ≥ 0.

Of course, conditions are needed to ensure that (1) has a zero solution. Lemma 2.4. [15, Proposition 1.9] For any r ≥ 1 and for arbitrary L02 -valued predictable process ϕ(·)

Z

sup E

s∈[0,t] r

where Cr = (r (2r − 1)) .

3

0

s

2r Z t   1 r

2r r ds , φ (u) dW (u) ≤ C E kφ (s)k 0 r L

0

2

t ≥ 0,

Main results In this section we state our main results.

Theorem 3.1. Assume that (H) holds. Let ρ = sup {tk − tk−1 } < ∞. Suppose that there exists a constant k∈N

0 < η < 1 satisfying |1 + ck | ≤ η, γ +

ln η ρ

< 0 and

 p  1−p ln η ln η M p−1 Υ =: γ + +3 −γ − Lpg ρ η ρ  p     (2−p)/2 M ln η p/2 + 3p−1 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) Lpσ < 0, η ρ

(2)

where 00 = 1. Then the zero solution of the system (1) is p-exponentially stable and the exponential convergence rate of (1) is greater than or equal to λ , where λ > 0 is the unique solution of  p  1−p M ln η ln η + λ + 3p−1 −γ − Lpg eλτ ρ η ρ  p     (2−p)/2 M ln η p/2 p−1 +3 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) Lpσ eλτ = 0. η ρ

γ+

Proof. Denote g (λ)

 p  1−p ln η M ln η + λ + 3p−1 −γ − Lpg eλτ ρ η ρ  p     (2−p)/2 M ln η p/2 p−1 +3 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) Lpσ eλτ . η ρ

= γ+

5

(3)

From (2), g(0) < 0 and g(+∞) = +∞. Since g (λ) is continuous and g ′ (λ) > 0 , Eq. (3) has a unique solution λ > 0. Furthermore, for any ε ∈ (0, λ), we have  p  1−p M ln η p−1 0 ≤ 3 −γ − Lpg e(λ−ε)τ η ρ  p     (2−p)/2 M ln η p/2 + 3p−1 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) Lpσ e(λ−ε)τ η ρ ln η < −γ − − (λ − ε) . ρ

(4)

From Lemma 2.2, we derive E kx (t)kpH

=

EkK (t, 0) x (0) + Z

Z

t

K (t, s) g (s, x (s − γ (s)))ds

0

t

K (t, s) σ (s, x (s − δ (s)))dW (s) kpH 0

Z t

p

p p−1

3 E kK (t, 0) x (0)kH + 3p−1 E K (t, s) g (s, x (s − γ (s)))ds

0 H

Z t

p

+ 3p−1 E K (t, s) σ (s, x (s − δ (s)))dW (s)

+

≤

0

=: 3p−1

3 X

H

Gi (t) .

(5)

i=1

We first evaluate the first term of the right-hand side. It follows from (i) of Lemma 2.3 that  p ln η M p p kϕkLp ep(γ+ ρ )t . G1 (t) = E kK (t, 0) x (0)kH ≤ η

(6)

As to the second term, by (H), the H¨ older inequality and (i) of Lemma 2.3, we obtain

Z t

p

G2 (t) = E K (t, s) g (s, x(s − γ(s)))ds

0

≤

≤

Z

H

p M (γ+ lnρη )(t−s) E e Lg |kx (s)kH |τ ds 0 η Z t  M p ln η 1−p p p (γ+ lnρη )(t−s) ( ) (−γ − ) Lg e |E kx (s)kH |τ ds . η ρ 0 t

(7)

As to the third term, by (H), the H¨ older inequality, (i) of Lemma 2.3 and Lemma 2.4, we obtain

Z t

p

G3 (t) = E K (t, s) σ (s, x(s − δ(s)))dW (s)

0

≤

= ≤

M p/2 ( )p (p (p − 1) /2) η (

Z t 

M p p/2 ) (p (p − 1) /2) η

0

Z

0

t

H

e

p(γ+ lnρη )(t−s)

2

2/p

ds

p/2

2/p p/2  ln η p e2(γ+ ρ )(t−s) E kσ (s, x(s − δ(s)))kL0 ds 2

Z t 2(γ+ ln η )(p−1) ρ M p p/2 (t−s) p−2 ) (p (p − 1) /2) e ds η 0 Z t ln η p × e(γ+ ρ )(t−s) E kσ (s, x(s − δ(s)))kL0 ds

(

!p/2−1

2

0

≤

p

E kσ (s, x(s − δ(s)))kL0

  (2−p)/2 M p ln η p/2 ( ) (p (p − 1) /2) 2(γ + ) (1 − p) / (p − 2) η ρ Z t  ln η × Lpσ e(γ+ ρ )(t−s) |E kx (s)kpH |τ ds . 0

6

(8)

It follows from (5)-(8) that p

E kx (t)kH

ln η M p p ) kϕkLp e(γ+ ρ )t η Z t  ln η ln η 1−p p M p ) Lg e(γ+ ρ )(t−s) |E kx (s)kH |τ ds + 3p−1 ( )p (−γ − η ρ 0   (2−p)/2 ln η p/2 p−1 M p 2(γ + ) (1 − p) / (p − 2) + 3 ( ) (p (p − 1) /2) η ρ Z t  ln η × Lpσ e(γ+ ρ )(t−s) |E kx (s)kpH |τ ds .

3p−1 (

≤

(9)

0

p

Without the loss of generality, we assume kϕkLp > 0. From M ≥ 1, 0 < η < 1 and λ > ε, it is easily observed that p



p

E kx (t)kH ≤ kϕkLp ≤ 3p−1

M η

p

p

kϕkLp e−(λ−ε)t ,

−τ ≤ t ≤ 0.

(10)

In the following, we shall prove that E

kx (t)kpH

<3

p−1



M η

p

kϕkpLp e−(λ−ε)t ,

t ≥ 0.

(11)

p

If this is not true, by (10) and the piecewise continuities of E kx (t)kH , then there must exist t∗ > 0 such that p

E kx (t)kH p

E kx (t)kH

p ∗ M p kϕkLp e−(λ−ε)t η  p M p p−1 < 3 kϕkLp e−(λ−ε)t , η ≥ 3p−1



(12) t < t∗ .

From (9), (13) and (4), we get p

E kx (t∗ )kH ≤ 3p−1



M η

p

kϕkLp e(γ+ p

ln η ρ

)t∗

! p  1−p Z t∗ ln η p γ+ lnρη )(t∗ −s) p ( +3 −γ − Lg e |E kx (s)kH |τ ds ρ 0  p     (2−p)/2 M ln η p/2 p−1 +3 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) η ρ ! Z t∗ ln η ∗ p × Lpσ e(γ+ ρ )(t −s) |E kx (s)kH |τ ds p−1

≤

3p−1





M η

0

M η 

p

kϕkLp e(γ+ p

ln η ρ

)t∗

p  1−p ln η +3 −γ − Lpg ρ !  p Z t∗ M p γ+ lnρη )(t∗ −s) p−1 −(λ−ε)(s−τ ) ( × e 3 kϕkLp e ds η 0  p     (2−p)/2 M ln η p/2 p−1 +3 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) η ρ !  p Z t∗ ln η ∗ M p (t −s) γ+ p p−1 −(λ−ε)(s−τ ) ρ ) × Lσ e( 3 kϕkLp e ds η 0 p−1

M η

7

(13)

≤

3

p−1

Z

× +3



< =

t∗



p kϕkLp

e(γ+

ρ

∗

)t

"

1+3

ln η e(−γ− ρ −(λ−ε))s e(λ−ε)τ ds

0

p−1

×Lpσ =

M η

Z

M η t∗

p

p/2

(p (p − 1) /2)

e(−γ−

ln η ρ −(λ−ε)

M η

   ln η −γ − ρ

−

Lpg

    (2−p)/2 ln η 2 γ+ (1 − p) / (p − 2) ρ !#

)s e(λ−ε)τ ds

0

M η 

p

!



"



p  1−p ln η 1+ 3 −γ − Lpg 3 ρ p     (2−p)/2 ln η M p/2 (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) + 3p−1 η ρ #  −1   ln η −γ− lnρη −(λ−ε))t∗ p (λ−ε)τ ( −γ − ×Lσ ) e − (λ − ε) e −1 ρ  p   ln η ln η ∗ ∗ M p p−1 kϕkLp e(γ+ ρ )t 1 + e(−γ− ρ −(λ−ε))t − 1 3 η  p ∗ M p 3p−1 kϕkLp e−(λ−ε)t . η p−1





p−1

p kϕkLp

γ+ lnρη

e(

∗

)t

p−1

M η

This contradicts (12), and so the estimate (11) holds. Letting ε → 0, we have  p M p p p−1 E kx (t)kH ≤ 3 kϕkLp e−(λ−ε)t , t ≥ 0. η The proof is complete.

According to Case (2) in Lemma 2.3, we have the following results for the case η ≥ 1. The proof is similar to the one in Theorem 3.1 and we omit it here. Theorem 3.2. Assume that (H) holds. Let θ = sup {tk − tk−1 } > 0. Suppose that there exists a constant k∈N

η ≥ 1 satisfying |1 + ck | ≤ η, γ +

ln η θ

< 0 and

 1−p ln η ln η p + 3p−1 (M η) −γ − Lpg θ θ     (2−p)/2 ln η + 3p−1 (M η)p (p (p − 1) /2)p/2 2 γ+ (1 − p) / (p − 2) Lpσ < 0. θ

Γ =: γ +

Then the zero solution of the system (1) is exponentially stable and the exponential convergence rate of (1) is greater than or equal to λ , where λ > 0 is the unique solution of  1−p ln η ln η p p−1 γ+ +λ+3 (M η) −γ − Lpg eλτ θ θ     (2−p)/2 ln η p p/2 p−1 +3 (M η) (p (p − 1) /2) 2 γ+ (1 − p) / (p − 2) Lpσ eλτ = 0. θ

8

4

Example To illustrate the efficiency of the results obtained, we will give two examples to which the existing results

cannot be applied. Example 4.1. Consider an impulsive control stochastic heat equation of the form  h 2 i ∂   dx (t, z) = ∂z 2 x (t, z) + 10x (t, z) + x (t − 1, z) dt       +x (t − 2, z) dB (t) , 0 < z < 1, t > 0, t = 6 tk  
∆x (tk , z) = − 21 x t− k , z , 0 < z < 1, t = tk ,      x (t, 0) = x (t, 1) , t ≥ 0,      x (s, z) = φ (s, z) ∈ P C b [−2, 0] , L2 [0, 1]
, −2 ≤ s ≤ 0, 0

(14)

F0

where B(t) is a standard Wiener process. Let H = L2 [0, 1] , H1 = W 2,2 (0, 1) ∩ W01,2 (0, 1) . Define operator

A by Ax = −



 ∂2 x (t, z) + 10x (t, z) , ∂z 2

x ∈ D(A) = H1 .

2 It can be verified that A generates a semigroup S(t) and kS (t)k ≤ e(10−π )t , t ≥ 0. Since γ = 10 − π 2 > 0,

the theorem 3.2 in [12] and theorem 3.5 in [13] are not available. Further, note that M = 1, τ = 2, Lg = 1, Lσ = 1 and η = |1 − 21 | =

1 2

< 1. Choose the impulsive control instant ρ = sup {tk − tk−1 } = 0.045. Let k∈N

p = 2, then by simply computation we get Υ < 0. So, it follows from Theorem 3.1 that the zero solution of (14) is mean square exponentially stable and the exponential convergence rate of (14) is greater than or equal to λ, where λ is the unique solution of  
ln 21 ln 12 10 − π 2 + + 12 −10 + π 2 − e2λ + 12e2λ = 0. 0.045 0.045

Example 4.2. Consider an impulsive control stochastic heat equation of the form  i h 2 ∂   dx (t, z) = ∂z 2 x (t, z) − 5x (t, z) + x (t − 1, z) dt       +x (t − 2, z) dB (t) , 0 < z < 1, t > 0, t = 6 tk  
∆x (tk , z) = x t− k , z , 0 < z < 1, t = tk ,      x (t, 0) = x (t, 1) , t ≥ 0,      x (s, z) = φ (s, z) ∈ P C b [−2, 0] , L2 [0, 1]
, −2 ≤ s ≤ 0, 0

(15)

F0

where B(t) is a standard Wiener process. Let H = L2 [0, 1] , H1 = W 2,2 (0, 1) ∩ W01,2 (0, 1) . Define operator A by Ax = −(

∂2 x (t, z) + 5x (t, z)), ∂z 2

x ∈ D(A) = H1 .

It can be verified that A generates a semigroup S(t) and kS (t)k ≤ e−(π

2

+5)t

, t ≥ 0. Note that γ = −(π 2 + 5),

M = 1, τ = 2, Lg = 1, Lσ = 1 and ρ = |1 + 1| = 2 > 1. Let p = 2, then   p/2 1−p/2 −γ 3p−1 M p Lpg γ 1−p + Lpσ (p(p − 1)/2) (2γ (p − 1) / (p − 2)) =

3 − π 2 − 5 < 0. π2 + 5

(16)

It follows from (16) and the theorem 3.1 in [5] that the corresponding continuous system is mean square exponential stability. Since η = 2 > 1, the theorem 3.2 in [12] and theorem 3.5 in [13] are not available. 9

Choose the impulsive control instant θ = inf {tk − tk−1 } = 0.5, then by simply computation we get Γ < 0. k∈N

So, it follows from Theorem 3.2 that the zero solution of (15) is mean square exponentially stable and the exponential convergence rate of (15) is greater than or equal to λ, where λ is the unique solution of  
ln 2 ln 2 2λ −5 − π 2 + + 12 5 + π 2 − e + 12e2λ = 0. 0.5 0.5

References [1] T.Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331(2007) 191-205. [2] K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50(2000) 273-278. [3] T. Caraballo, J. Real, T. Taniguchi, The exponential stability of neutral stochastic delay partial differential equations, Discrete Contin. Dyn. Syst. 18(2007) 295-313. [4] T. Caraballo, K. Liu. Exponential stability of mild solutions of stochastic partial differential equations with delays. Stoch. Anal. Appl. 17(1999) 743-763. [5] J.W. Luo, Fixed points and exponential stability of mild solutions of stochastic partial differential equations with delays , J. Math. Anal. Appl. 342 (2008) 753-760. [6] J.W. Luo. Fixed points and stability of neutral stochastic delay differential equations. J. Math. Anal. Appl. 334(2007) 431-440. [7] D.S. Li, D.Y. Xu, Attracting and quasi-invariant sets of stochastic neutral partial functional differential equations, Acta Math. Scientia 33(2013) 578-588. [8] R. Sakthivel, Y. Ren, H, Kim, Asymptotic stability of second-order neutral stochastic differential equations, J. Math. Phy. 51(2010) no.5. [9] R. Sakthivel, Y. Ren, Exponential stability of second-order stochastic evolution equations with Poisson jumps, Commun. Nonlinear Sci. Numer. Simulat. 17(2012) 4517-4523. [10] X.H. Wang, Z.G. Yang, The domain of attraction and the stability region for stochastic partial differential equations with delays, Nonl. Anal. 75 (2012) 6465-6472. [11] S.J. Long, L.Y. Teng, D.Y. Xu, Global attracting set and stability of stochastic neutral partial functional differential equations with impulses, Statist.Probab. Lett. 82 (2012) 1699-1709. [12] H.B. Chen, Impulsive-integral inequality and exponential stability for stochastic partial differential equations with delays, Statist. Probab. Lett. 80 (2010) 50-56. [13] R. Sakthivel, J.W. Luo, Asymptotic stability of impulsive stochastic partial differential equations with infinite delays, J. Math. Anal. Appl. 356 (2009) 1-6. 10

[14] G. Da Prato, J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. [15] A.Ichikawa, Stability of semilinear stochastic evolution equations. J. Math. Anal. Appl. 90 (1) 1982, 12-44. [16] D.Y. Xu, B. Li, S.J. Long, L.Y. Teng, Moment estimate and existence for solutions of stochastic functional differential equations, Nonlinear Anal. 108(2014) 128-143.

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