Stability analysis for second order stochastic neutral partial functional systems subject to infinite delays and impulses

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R E S E A R C H

Open Access

Stability analysis for second-order

stochastic neutral partial functional systems

subject to inﬁnite delays and impulses

Feng Jiang

1*

_{, Hua Yang}

2,3

_{and Tianhai Tian}

4

*_{Correspondence:}

[email protected]

1_{School of Statistics and}

Mathematics, Zhongnan University of Economics and Law, Hubei, 430073, China

Full list of author information is available at the end of the article

Abstract

The paper is concerned with stability for second-order stochastic neutral partial functional systems subject to inﬁnite delays and impulses. Some suﬃcient conditions ensuringpth moment exponential stability of the second-order stochastic systems are given by gaining a new integral inequality with impulses. Some earlier results are generalized and improved.

1 Introduction

The role played by second-order system in the dynamics of physical and biological systems has been an issue of growing interest and discussion. Recent research has also focused on the relative dominance of existence, uniqueness, and stability behavior of second-order systems [–].

In the real world, noise plays an important role in comprehending phenomena that can be very hard to describe by deterministic systems. The dynamical behavior of stochas-tic partial differential systems (SPDSs) has been discussed [–]. Sakthivelet al.[, ] studied pseudo almost automorphic mild solutions to stochastic fractional equations by a fixed point strategy and discussed complete controllability of stochastic evolution equa-tions with jumps, respectively. Recently, [, ] discussed the existence, uniqueness, and stability of mild solutions to stochastic neutral partial functional systems (SNPFSs). Mean-while, second-order SPDSs [] are often used to model charge on a condenser, mechani-cal vibrations, etc. Liang and Guo [] discussed the asymptotic behavior for second-order stochastic evolution equations with memory. Sakthivel and Ren [] studied exponential stability of second-order stochastic evolution equations with Poisson jumps by fixed point theory. In addition, since sudden variations often exist in practice, it is necessary to model the noise source by a sequence of impulses. For instance, the stability for first-order impul-sive SPDSs was discussed by using integral inequalities in [–]. The authors in [–] discussed the existence, controllability, and stability of mild solutions of second-order im-pulsive stochastic systems.

The papers mentioned above focus on stochastic systems with finite delays. Recently, the study of stochastic systems with infinite delay [–] has also been reported. For ex-ample, Ren and Sakthivel [] discussed existence, uniqueness, and stability of a mild solu-tion for second-order neutral stochastic evolusolu-tion systems with infinite delay and Poisson

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jumps by fixed point theory. Yue [] discussed the existence of mild solutions of second-order neutral impulsive stochastic evolution systems with infinite delay by contraction theory. The authors in [, ] extended second-order stochastic systems with infinite delay to second-order jump systems and second-order integro-differential systems.

With motivation from the above discussions, in order to overcome the difficulty of im-pulses and infinite delays, in this paper we first establish a new integral inequality with re-spect to impulses and infinite delays, which makes an important contribution to stability analysis of the second-order SNPFSs subject to infinite delays and impulses. The integral inequality is new and different from the above papers since the impulses and infinite de-lays exist in the system. The inequality with finite dede-lays in [–, ] is extended to the inequality with infinite delays and impulses such that it is effective for the stochastic sys-tems with infinite delays and impulses. By the new inequality, together with the stochastic analysis technique, some sufficient conditions for stability are obtained.

The rest of this paper is organized as follows. In Section , some preliminaries are intro-duced. Section  presents stability results by establishing a new integral inequality with impulses and inﬁnite delays and stochastic analysis technique. In Section , we give an example to show the eﬀectiveness of the obtained results.

2 Preliminaries

In this paper, (,F,{Ft}t≥,P) is a complete probability space equipped with some

ﬁl-trationFt (t≥) satisfying the usual conditions, i.e., the ﬁltration is right continuous

andFcontains allF-null sets. Let and be two real separable Hilbert spaces and

L(,) be the space of bounded linear operators fromto. · denotes the norms

in,, andL(,).C((–∞, ],) is the space of all bounded and continuous

func-tionsϕfrom (–∞, ] towith the norm · C=sup–∞<θ≤ϕ(θ).Jis the family of all Ft(t≥)-measurable andC((–∞, ],)-valued random variables. Let the stochastic

pro-cessw(t) be aQ-Wiener process andσ∈L(,) be aQ-Hilbert-Schmidt operator with

σ L =

tr(σQσ∗) < +∞.L_(,) denotes the space of allQ-Hilbert- Schmidt operators

σ:→. For details, we can refer to [] and the references therein.

Consider the following second-order stochastic neutral partial functional systems (SNPFSs) subject to inﬁnite delays and impulses:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

d[y(t) –μ(t,yt)] = [Ay(t) +f(t,yt)]dt+g(t,yt)dw(t), t≥,t=tj,j= , , . . . , y(tj) =Ij(y(t–j)), j= , , . . . ,

y(tj) =Ij(y(tj–)), j= , , . . . ,

y(·) =ϕ∈J, y() =,

()

whereis also anF-measurable-valued random variable independent of the Wiener processw(t).A:D(A)⊂→is the inﬁnitesimal generator of a strongly continuous cosine family on;f,μ: [, +∞)×J→,g: [, +∞)×J→L

(,),yt: (–∞, ]→ ,yt(θ) =y(t+θ) (t≥),  <t<t<· · ·<tj<· · ·, andlimj→+∞tj= +∞.Ij,Ij:J→H,

ξ(t) =ξ(t+_{) –}_ξ₍_t–_{), where}_ξ₍_t+_{) and}_ξ₍_t–_{) denote the right and left limits of}_ξ _at_t_, re-spectively.

Let{C(t) :t∈R} ⊂L(,) be a strongly continuous cosine family (see [] and the

references therein) and the corresponding strongly continuous sine family{S(t) :t∈R} ⊂ L(,) be deﬁned by S(t)y=_tC(s)y ds,t∈R,y∈. The generatorA:→ of

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Lemma ([]) Let A be the inﬁnitesimal generator of a cosine family of operators{C(t) : t∈R}.Then:

(i) There existδ∗≥andβ≥such thatC(t) ≤δ∗eβt_{and hence}_S₍_t₎_≤_δ∗_eβt_.

(ii) For≤s≤r< +∞,A_srS(u)y du= [C(r) –C(u)]y.

(iii) There existsδ∗∗≥such that for≤r≤s< +∞ S(s) –S(r) ≤δ∗∗|_rseβ|θ|_d_θ_|_.

Lemma ([]) For any p≥and for an L(,)-valued predictable process y(·)we have

sup

s∈[,t] E

s



y(v)dw(v)

p

≤cp t



Ey(v)p

L 

p_dv

p/

, t∈[, +∞),

where cp= (p(p– )/)p/.

Deﬁnition  An-value stochastic processy(t) (t∈R) is called a mild solution of sys-tem (), if

(i) y(t)is adapted toFt(t≥) and has a càdlàg path ont≥almost surely.

(ii) Fort∈[, +∞), almost surely

y(t) =C(t)ϕ+S(t)ξ–μ(,ϕ)+

t



C(t–ν)μ(ν,yν)dν

+

t



S(t–ν)f(ν,yν)dν+ t



S(t–ν)g(ν,yν)dw(ν)

+

<tj<t

C(t–tj)Ij

yt_j–+ <tj<t

S(t–tj)Ij

yt_j–. ()

Deﬁnition  We say there is a mild solution of system (), or for simplicity system ()

is said to bepth (p≥) moment exponentially stable, if there exist constantsγ >  and N>  such that

Ey(t)p≤Ne–γt, t≥,p≥,

for the initial dataϕ∈J.

3 Main results

In the section, we will establish exponential stability of system () by some integral in-equalities. Thus we need to employ the following assumptions:

(A) For some constantsα≥,a> , andb> ,C(t) ≤αe–bt_and_S₍_t₎_≤_α_e–at_,_t_≥_.

(A) There exist constants Li> (i= , , ) and a functionκ : (–∞, ]→[, +∞)with



–∞κ(t)dt= and

 –∞κ(t)e

–t_dt_{< +}_∞_for_{> , such that}

μ(t,y) –μ(t,y)≤L 

–∞

κ(ϑ)y(t+ϑ) –y(t+ϑ)dϑ,

f(t,y) –f(t,y)≤L 

–∞κ(ϑ)

y(t+ϑ) –y(t+ϑ)dϑ,

g(t,y) –g(t,y)L_≤L



–∞

κ(ϑ)y(t+ϑ) –y(t+ϑ)dϑ,

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(A) Fory,y∈ and

+∞

j= αj< +∞, +∞

j= βj< +∞, there exist positive numbersαj,βj

(j= , , . . .) such that

Ij(y) –Ij(y)≤αjy–y,

Ij(y) –Ij(y)≤βjy–y,

andIj() =Ij() = .

Remark  References [, ] discussed the existence of mild solutions of second-order

stochastic neutral partial functional equations with inﬁnite delay or with ﬁnite delay, re-spectively. Similarly, we can show that for system () there exists a unique mild solution under the conditions (A)-(A). Also, it is obvious that system () has a unique trivial mild solution when the initial valueϕ= .

To obtain exponential stability of system (), we ﬁrst establish the following integral in-equality with impulses and inﬁnite delays.

Lemma  Suppose thatη,η∈(,]and there exist constantsζi>  (i= , , , )and a

function:R→[, +∞)such thatζ_η+ζ_η ++_j_=∞(aj+bj) < ,and

(t)≤

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

ζe–ηt+ζe–ηt+ζ

t

e–

η(t–s)

–∞κ(ϑ)(s+ϑ)dϑds +ζ

t

e–

η(t–s)

–∞κ(ϑ)(s+ϑ)dϑds

+_t j<taje

–η(t–tj)₍_t–

j) +

tj<tbje

–η(t–tj)₍_t–

j), t≥, ζe–ηt+ζe–ηt, t∈(–∞, ],

()

holds. Then (t)≤Ne–δt,t ∈(–∞, +∞), whereδ ∈(,η ∧η) is a root of the inte-gral equation: ( ζ

η–δ + ζ η–δ)

 –∞κ(ϑ)e

–δϑ_d_ϑ₊+∞

j=(aj+bj) =  and N=max{ζ+ζ,

ζ(η–δ)

ζ



–∞κ(ϑ)e–δϑdϑ,

ζ(η–δ)

ζ



–∞κ(ϑ)e–δϑdϑ}> .

Proof LetF(ζ) = ( ζ η–ζ +

ζ η–ζ)



–∞κ(ϑ)e–ζ ϑdϑ+ +∞

j=(aj+bj) – , then it is obvious that

there exists a positive constantδ∈(,η∧η), such thatF(δ) = . For anyε>  and let

Nε=max

ζ+ζ+ε,

(η–δ)(ζ+ε)

ζ



–∞κ(ϑ)e–δϑdϑ

, (η–δ)(ζ+ε)

ζ



–∞κ(ϑ)e–δϑdϑ

> . ()

Now we only need to claim that () implies

(t)≤Nεe–δt, t∈(–∞, +∞). ()

Obviously, fort∈(–∞, ], () holds. Next we will prove () by the contradiction method. Assume that there exists at>  such that

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Note thatδ∈(,η∧η), from () we have

Remark  Lemma , which is diﬀerent from the lemmas of [, , ], plays an important

role here. The main lemmas in [, , ] cannot be applied in the present study because of the effects of impulses and infinite delays. The main lemma of [] aims at first-order stochastic systems, while that of [] focuses on finite constant delay. Our lemma is effec-tive to establish the stability of system (). It is clear that the lemma can deal with impulses and infinite delay terms.

Theorem  Assume that(A)-(A)hold and a,b∈(,],and for p≥,

Then system()is pth moment exponentially stable.Especially,when p= system()is mean square exponentially stable providedα_L

b–+αLa–+αLa–+α(

+∞

j= βj)+ α₍+∞

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≤αp

From Lemma , we obtain

E

These together with () yield

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+∞

j= η˘+

+∞

j= ηˇ< ,i.e. () holds, then by Lemma , we obtainEy(t) ≤ ˆMe–

ˆ δt_,_t_∈

[, +∞), whereδˆ∈(,η∧η) andδˆis a positive root of the equation (_bζ˘_–_δ+_aζ˘_–_δ)



–∞κ(ϑ)×

e–δϑ_d_ϑ ₊ +∞

j=(αj + βj) = , Mˆ = {p–αp(Eϕp + E – μ(,ϕ)p), p–αpb–pLp, p–_αp₍_a–p_Lp

+cpLpa– p

((p–) p– )

–p_₎_}_{> . The proof is completed.}

IfD(t,yt)≡, system () becomes the following second-order SPFDSs subject to inﬁnite

delays and being impulsive:

⎧ ⎪ ⎨ ⎪ ⎩

dy(t) = [Ay(t) +f(t,yt)]dt+g(t,yt)dw(t), t≥,t=tj,j= , , . . . , y(tj) =Ij(y(t–j)), y(tj) =Ij(y(tj–)), j= , , . . . ,

y(·) =ϕ∈J, y() =,

()

Corollary  Under(A)-(A),and a,b∈(,],system()is pth(p≥)moment exponen-tially stable provided

p–αpLp_a–p+ p–αpLp_a–p

p(p– ) 

p

_(_p_{– )}

p– 

–p

+ p–αp ₊_∞

j=

βj p

+ p–αp ₊_∞

j=

αj p

< . ()

Moreover,if p=  system()is mean square exponentially stable providedαL a–+ α_L

a–+ α(

+∞

j= βj)+ α( +∞

j= αj)< .

If system () is not subject to impulses, system () becomes the following second-order SNPFSs:

d[y(t) –μ(t,yt)] = [Ay(t) +f(t,yt)]dt+g(t,yt)dw(t), t≥,t=tj,j= , , . . . ,

y(·) =ϕ∈J, y() =,

()

Corollary  Under(A)-(A),and a,b∈(,],system()is pth(p≥)moment expo-nentially stable provided

p–αpLp_b–p+ p–αpLp_a–p+ p–αpLp_a–p

p(p– ) 

p

_(_p_{– )}

p– 

–p_

< . ()

Moreover,if p= system()is mean square exponentially stable provided αL b–+ αL_a–+ αL_a–< .

Remark  System () with ﬁnite delays or without impulses has been discussed in [,

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generalization and development of these existing results. Finally, it should be pointed out that the results are valid for system () with ﬁnite delays. Our results can easily be extended to second-order stochastic systems driven by a Lévy process.

Remark  If p–_αp_Lp

tral term is alsopth moment exponentially stable. IfαL

b–+ αLa–+αLa–< /, system () without neutral term is mean square exponentially stable.

4 An example

In this section, we will discuss an example to show the theoretical results. Consider the following second-order stochastic neutral partial functional systems subject to inﬁnite de-lays and impulses:

pth moment exponentially stable provided

up+u

Especially, whenp= , system () is mean square exponentially stable providedu +u+ the second-order neutral stochastic systems with inﬁnite delays. Now letα= ,b=a=

π_,_L

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pth moment exponentially stable provided

up_+up_+up_

p(p– ) 

p

_(_p_{– )}

p– 

–p_ < ;

and ifu

+u+u< , system () without impulses is mean square exponentially stable.

Remark  It is clear that the algebraic conditions of stability in the example are easily

computed. The example also shows that the results of this paper are eﬀective.

5 Conclusions

In this paper we have studied second-order stochastic neutral partial functional systems subject to infinite delays and impulses. We first established a new integral inequality with respect to impulses and then give some algebraic criteria to ensurepth moment expo-nential stability of the second-order stochastic impulsive systems. The results show some earlier results are generalized and improved. Finally, an example was given to illustrate the effectiveness of the theoretical results. Further research will be to study controller design and noise stabilization as well as its possible extension to second-order stochastic systems driven by Lévy jumps.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

FJ and HY jointly carried out the study of stability of this paper and drafted the manuscript. TT provided some constructive suggestions for the improvement of this paper. All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Statistics and Mathematics, Zhongnan University of Economics and Law, Hubei, 430073, China.}2_{School of}

Mathematics & Computer Science, Wuhan Polytechnic University, Hubei, 430023, China.3_{School of Automation,}

Huazhong University of Science and Technology, Hubei, 430074, China. 4_{School of Mathematical Sciences, Monash}

University, Melbourne, VIC 3800, Australia.

Acknowledgements

The work is supported by the National Natural Science Foundation of China under Grant No. 61304067 and 11571368, the Natural Science Foundation of Hubei Province of China under Grant No. 2013CFB443, the Science and Technology Research Project of Hubei Provincial Department of Education under Grant No. Q20151705 and the Australian Research Council Future Fellowship under Grant No. FT100100748.

Received: 6 October 2015 Accepted: 23 August 2016

References

1. Tsai, C, Chen, H, Hsu, JRC: Second-order time-dependent mild-slope equation for wave transformation. Math. Probl. Eng.2014, 341385 (2014)

2. Da Prato, G, Zabczyk, J: Second-Order Partial Diﬀerential Equations in Hilbert Spaces. Cambridge University Press, Cambridge (2002)

3. Feintuch, A: Asymptotic behaviour of inﬁnite chains of coupled kinematic points: second-order equations. Math. Control Signals Syst.26(3), 463-480 (2014)

4. Biles, D, Spraker, J: Uniqueness of solutions for second order diﬀerential equations. Monatshefte Math.178(2), 165-169 (2015)

5. Travis, CC, Webb, GF: Cosine families and abstract nonlinear second order diﬀerential equations. Acta Math. Acad. Sci. Hung.32, 76-96 (1978)

6. Da, PG, Zabczyk, J: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) 7. Liu, L: Stability of Infinite Dimensional Stochastic Differential Equations with Applications. Chapman & Hall/CRC,

London (2006)

8. Mckibben, MA: Second-order neutral stochastic evolution equations with heredity. J. Appl. Math. Stoch. Anal.2, 177-192 (2004)

9. Luo, J: Fixed points and exponential stability of mild solutions of stochastic partial diﬀerential equation with delays. J. Math. Anal. Appl.342, 753-760 (2008)

(11)

11. Bao, J, Hou, Z, Yuan, C: Stability in distribution of mild solutions to stochastic partial diﬀerential equations. Proc. Am. Math. Soc.138, 2169-2180 (2010)

12. Sakthivel, R, Revathi, P, Marshal Anthoni, S: Existence of pseudo almost automorphic mild solutions to stochastic fractional diﬀerential equations. Nonlinear Anal., Theory Methods Appl.75, 3339-3347 (2012)

13. Sakthivel, R, Ren, Y: Complete controllability of stochastic evolution equations with jumps. Rep. Math. Phys.68, 163-174 (2011)

14. Jiang, J, Shen, Y: A note on the existence and uniqueness of mild solutions to neutral stochastic partial functional diﬀerential equations with non-Lipschitz coeﬃcients. Comput. Math. Appl.61, 1590-1594 (2011)

15. Yang, X, Zhu, Q:pth moment exponential stability of stochastic partial diﬀerential equations with Poisson jumps. Asian J. Control16, 1482-1491 (2014)

16. Arora, U, Sukavanam, N: Approximate controllability of second order semilinear stochastic system with nonlocal conditions. Appl. Math. Comput.258, 111-119 (2015)

17. Liang, F, Guo, Z: Asymptotic behavior for second order stochastic evolution equations with memory. J. Math. Anal. Appl.419(2), 1333-1350 (2014)

18. Sakthivel, R, Ren, Y: Exponential stability of second-order stochastic evolution equations with Poisson jumps. Commun. Nonlinear Sci. Numer. Simul.17, 4517-4523 (2012)

19. Chen, H: Impulsive-integral inequality and exponential stability for stochastic partial diﬀerential equations with delays. Stat. Probab. Lett.80, 50-56 (2010)

20. Long, S, Teng, L, Xu, D: Global attracting set and stability of stochastic neutral partial functional diﬀerential equations with impulses. Stat. Probab. Lett.82, 1699-1709 (2012)

21. Yang, H, Jiang, F: Exponential stability of mild solutions to impulsive stochastic neutral partial diﬀerential equations with memory. Adv. Diﬀer. Equ.2013, 148 (2013)

22. Ren, Y, Dandan, S: Second-order neutral impulsive stochastic evolution equations with delay. J. Math. Phys.50, 102709 (2009)

23. Arthi, G, Balachandran, K: Controllability of damped second order impulsive neutral integrodiﬀerential systems with nonlocal conditions. J. Control Theory Appl.11, 186-192 (2013)

24. Arthi, G, Park, JH, Jung, HY: Exponential stability of second-order neutral stochastic diﬀerential equations with impulses. Int. J. Control88(6), 1300-1309 (2015)

25. Luo, J: Stability of stochastic partial diﬀerential equations with inﬁnite delays. J. Comput. Appl. Math.222, 364-371 (2008)

26. Ren, Y, Hu, L, Sakthivel, R: Controllability of impulsive neutral stochastic functional diﬀerential inclusions with inﬁnite delay. J. Comput. Appl. Math.235, 2603-2614 (2011)

27. Sakthivel, R, Luo, J: Asymptotic stability of impulsive stochastic partial diﬀerential equations with inﬁnite delays. J. Math. Anal. Appl.356, 1-6 (2009)

28. Jiang, F, Shen, Y: Stability of impulsive stochastic neutral partial diﬀerential equations with inﬁnite delays. Asian J. Control14, 1706-1709 (2012)

29. Chen, H: The asymptotic behavior for second-order neutral stochastic partial diﬀerential equations with inﬁnite delay. Discrete Dyn. Nat. Soc.2011, 584510 (2011)

30. Ren, Y, Sakthivel, R: Existence, uniqueness, and stability of mild solution for second-order neutral stochastic evolution equations with inﬁnite delay and Poisson jumps. J. Math. Phys.53, 073517 (2012)

31. Yue, C: Second-order neutral impulsive stochastic evolution equations with inﬁnite delay. Adv. Diﬀer. Equ.2014, 112 (2014)

32. Palanisamy, M, Chinnathambi, R: Approximate controllability of second-order neutral stochastic diﬀerential equations with inﬁnite delay and Poisson jumps. J. Syst. Sci. Complex.28(5), 1033-1048 (2015)