A delay-dependent stability criterion for nonlinear stochastic delay-integro-differential equations

Acta Mathematica Scientia 2011,31B(5):1813–1822 http://actams.wipm.ac.cn

A DELAY-DEPENDENT STABILITY CRITERION FOR NONLINEAR STOCHASTIC DELAY-INTEGRO-DIFFERENTIAL EQUATIONS∗

)1,2

Niu Yuanling (

 )1†

Zhang Chengjian (


)3

Duan Jinqiao (

1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China 2. School of Mathematics, Central South University, Changsha 410075, China 3. Department of Applied Mathematics, Illinois Institute of Technology, Chicago IL 60616, USA E-mail: [email protected]; [email protected]; [email protected]

Abstract A type of complex systems under both random influence and memory effects is considered. The systems are modeled by a class of nonlinear stochastic delay-integrodifferential equations. A delay-dependent stability criterion for such equations is derived under the condition that the time lags are small enough. Numerical simulations are presented to illustrate the theoretical result. Key words complex systems under uncertainty; mean-square exponential stability; stochastic delay-integro-differential equations; memory effects; numerical experiment 2000 MR Subject Classification

1

65C30; 34K20; 60H35

Introduction

Random fluctuations are abundant in natural or engineered systems. Therefore, stochastic modelling has come to play an important role in various fields such as biology, mechanics, economics, medicine and engineering (see [1–5]). Moreover, these systems are sometimes subject to memory effects, when their time evolution depends on their past history with noise disturbance. Stochastic delay differential equations (SDDEs) are often used to model such systems. They can be regarded as generalizations of both deterministic delay differential equations (DDEs) and stochastic ordinary delay differential equations (SODEs). The stability analysis is very important for delay systems as we like to know the impact of memory as well as noise. This motivates a lot of recent researches; see, for example, [1, 6–13] and the references therein. However, most of the criteria obtained are independent of time delay. Such criteria are in general good for large delay but might not be sufficient for small delay. ∗ Received

March 19, 2010; revised August 27, 2010. This work is supported by NSFC (10871078), 863 Program of China (2009AA044501), an Open Research Grant of the State Key Laboratory for Nonlinear Mechanics of CAS, and Graduates’ Innovation Fund of HUST (HF-08-02-2011-011). † Corresponding author: Zhang Chengjian.

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In [14], Mao and Shaikhet considered the following nonlinear SDDE with Markovian switching dx(t) = f (x(t), x(t − τ1 ), t, r(t))dt + g(x(t), x(t − τ2 ), t, r(t))dw(t),

(1.1)

as a perturbed system of the corresponding nonlinear jump equation dx(t) = f (x(t), x(t), t, r(t))dt.

(1.2)

It was proved that (1.1) is stable whenever (1.2) is stable, as long as the time lag τ1 and the stochastic perturbation g(x(t), x(t − τ2 ), t, r(t))dw(t) are both sufficiently small. Such delaydependent stability criteria are very useful for systems with small delay. So far, as we know, the above analytical method was only applied to SDDEs with discrete delays. If the analytical method is applied to the SDDEs with both discrete and distributed delays, that is, SDIDEs, what stability criterion can be obtained? In the present paper, we will focus on this issue. The following SDIDE  dx(t) = f (t, x(t), x(t − τ1 ), +g(t, x(t), x(t − τ3 ),

t t−τ2  t

h1 (t, s, x(s))ds)dt

t−τ4

h2 (t, s, x(s))ds)dw(t)

(1.3)

can be regarded as a perturbed system of the nonlinear differential equation dx(t) = f (t, x(t), x(t), 0)dt.

(1.4)

This paper is organized as follows. In Section 2, we show that, under suitable conditions, the nonlinear SDIDE (1.3) is exponentially stable in mean-square sense if the corresponding ODE (1.4) is exponentially stable. In Section 3, an example with numerical experiments is presented to illustrate the effectiveness of the proposed method.

2

Exponential Stability of Nonlinear SDIDEs

Let w(t) = (w1 (t), · · · , wm (t))T be an m-dimensional Wiener process defined on a filtered probability space (Ω, A, P). Denote by C([−τ, 0]; Rn ) the Banach space consisting of all continuous paths from [−τ, 0] to Rn , equipped with the norm μ = sup |μ(s)|. s∈[−τ,0]

Consider the following Itˆ o-type scalar SDIDEs with delays τi > 0, i = 1, 2, 3, 4:  t ⎧ ⎪ ⎪ dx(t) = f (t, x(t), x(t − τ ), h1 (t, s, x(s))ds)dt 1 ⎪ ⎪ ⎪ t−τ2 ⎪ ⎨  t

h2 (t, s, x(s))ds)dw(t), t ≥ 0, +g(t, x(t), x(t − τ3 ), ⎪ ⎪ ⎪ t−τ4 ⎪ ⎪ ⎪ ⎩ x(t) = ψ(t), t ∈ [−τ, 0], τ = max {τi },

(2.1)

i=1,2,3,4

where ψ(t) is an A0 -measurable C([−τ, 0]; R)-valued random variable with Eψ2 < ∞. The functions f : R+ × Rn × Rn × Rn → Rn , g : R+ × Rn × Rn × Rn → Rn×m , h1 : R+ × [−τ2 , +∞) ×

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Rn → Rn and h2 : R+ × [−τ4 , +∞) × Rn → Rn are assumed to be continuous and satisfy the conditions f (t, x, x, x) − f (t, x, y, y˜)2 ≤ k1 x − y2 + k2 x − y˜2 ,

(2.2)

f (t, x, y, y˜)2 ≤ k3 x2 + k4 y2 + k5 ˜ y 2 ,

(2.3)

2xT f (t, x, x, x) ≤ αx2 ,

(2.4)

g(t, x, z, z˜)2 ≤ βx2 + γz2 + δ˜ z 2 ,

(2.5)

h1 (t, s, x)2 ≤ k6 x2 ,

(2.6)

h2 (t, s, x)2 ≤ k7 x2 ,

(2.7)

in which α ∈ R, β, γ, δ, ki (i = 1, 2, · · · , 7) all are positive constants, and  ·  denotes a given norm in Rn . Note that from these hypotheses we have f (t, 0, 0, 0) ≡ 0 and g(t, 0, 0, 0) ≡ 0. Moreover, we assume that h1 (t, s, 0) ≡ 0 and h2 (t, s, 0) ≡ 0. So with the trivial initial function ψ(t) = 0 for t ∈ [−τ, 0], equation (2.1) admits a trivial solution x(t, 0) ≡ 0. b Besides, we shall need a few more notations. Denote by CA ([−τ, 0]; Rn ) the family of 0 n all bounded, A0 -measurable, C([−τ, 0]; R )-valued random variables. Provided that x(t) is a continuous Rn -valued stochastic process on t ∈ [−τ, ∞), let xt = x(t + s) : −τ ≤ s ≤ 0 for t ≥ 0, which is a C([−τ, 0]; Rn )-valued stochastic process. Denote by C 1,2 (R+ × Rn ; R+ ) the family of all nonnegative functions V (t, x) on R+ × Rn , which have first order continuous derivative with respect to t and second order continuous derivatives with respect to x. For each function V ∈ C 1,2 (R+ ×Rn ; R+ ) the operator LV associated with (2.1) from R+ ×Rn ×Rn ×Rn ×Rn ×Rn to R is defined by the formula LV (t, x, y, y˜, z, z˜) = Vt (t, x) + Vx (t, x)f (t, x, y, y˜) 1 + trace[g T (t, x, z, z˜)Vxx (t, x)g(t, x, z, z˜)], 2

(2.8)

where Vt (t, x) = ∂V (t, x)/∂t,

(2.9)

Vx (t, x) = (∂V (t, x)/∂x1 , · · · , ∂V (t, x)/∂xn ),

(2.10)

Vxx (t, x) = (∂ 2 V (t, x)/∂xi ∂xj )n×n .

(2.11)

and

Definition 1

System (2.1) is called exponentially stable in mean-square if lim sup

t→∞

1 log(Ex(t; ξ)2 ) < 0 t

(2.12)

b for initial function ξ ∈ CA . 0 Based on the idea in [13], where stability of nonlinear SDDEs without distributed delay was concerned, we derive a mean-square exponential stability result of (2.1) as follows. Theorem 1 Assume that SDIDE (2.1) satisfies conditions (2.2)–(2.7) and

α+β+γ+2

 2k2 < 0.

(2.13)

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Then (2.1) is exponentially stable in mean-square whenever τ˜ < σ, with σ > 0 being the unique positive root of the equation C(σ) = 0. (2.14) Here τ˜ = max {τi }, i=1,2,4

C(σ) = 4k1 k5 k6 θ−1 σ 4 + 4k1 k7 δθ−1 σ 3 + 2k1 (k3 + k4 )θ−1 σ 2 + (2(k1 (β + γ) +k2 k6 )θ−1 + δk7 )σ + α + β + γ + θ + 2θ−1 k2 ,

(2.15)

and θ > 0 satisfies α + β + γ + θ + 2θ−1 k2 < 0.

(2.16)

b . For some ε > 0 small Proof Write x(t) = x(t, ξ) and fix any initial function ξ ∈ CA 0 enough, define V (t, x) = exp(εt)x2 , (t, x) ∈ R+ × Rn . (2.17)

Obviously, V ∈ C 1,2 (R+ × Rn ; R+ ).

(2.18)

LV (t, x, y, y˜, z, z˜) = exp(εt)[εx2 + 2xT f (t, x, y, y˜) + g(t, x, z, z˜)2 ].

(2.19)

Then Moreover, (2.13) implies that the positive solution θ exists for inequality (2.16). Thus, with (2.2), (2.4), (2.5) and the elementary inequality 2ab ≤ θa2 + θ−1 b2 , we obtain 2xT f (t, x, y, y˜) = 2xT f (t, x, x, x) + 2xT [f (t, x, y, y˜) − f (t, x, x, x)] ≤ αx2 + θx2 + θ−1 f (t, x, y, y˜) − f (t, x, x, x)2 ≤ (α + θ)x2 + θ−1 (k1 x − y2 + k2 x − y˜2 )

(2.20)

and z 2 . g(t, x, z, z˜)2 ≤ βx2 + γz2 + δ˜

(2.21)

Substituting (2.20) and (2.21) into (2.19) gives z 2 LV (t, x, y, y˜, z, z˜) ≤ exp(εt)[(ε + α + β + θ)x2 + γz2 + δ˜ +θ−1 k1 x − y2 + θ−1 k2 x − y˜2 )].

(2.22)

It is clear from the definition of V (t, x) that EV (0, x(0)) = Ex(0)2 ≤ Eξ2 := M1 .

(2.23)

Thus, applying the Itˆ o formula, we conclude that exp(εt)EV (t, x(t)) ≤ M1 + (α + β + θ + 2ε)



t

exp(2εs)x(s)2 ds + γ

0

 0

t

exp(2εs)x(s − τ3 )2 ds

 s 2    +δ exp(2εs)E  h2 (s, r, x(r))dr   ds 0 s−τ4  t +θ−1 k1 exp(2εs)x(s) − x(s − τ1 )2 ds 0  t  s exp(2εs)Ex(s) − h1 (s, r, x(r))dr2 ds. +θ−1 k2 

t

0

s−τ2

(2.24)

No.5

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Next, we always let t ≥ τ . Therefore  t exp(2εs)x(s − τ3 )2 ds 0

 ≤

t−τ3

−τ3  0

= −τ3

exp(2ε(s + τ3 ))x(s)2 d(s + τ3 ) 

exp(2ε(s + τ3 ))Ex(s)2 ds +

≤ τ3 exp(2ετ3 )Eξ2 + exp(2ετ3 )



t

t−τ3

0

exp(2ε(s + τ3 ))Ex(s)2 ds

exp(2εs)Ex(s)2 ds.

(2.25)

0

Furthermore, it follows from (2.1) that  t Ex(t) − x(t − τ1 )2 ≤ 2τ1 E

 2  s   f (s, x(s), x(s − τ1 ),  ds h (s, r, x(r))dr) 1   t−τ1 s−τ2    t  s  2 g(s, x(s), x(s − τ3 ),  ds. (2.26) +2E h (s, r, x(r))dr) 2   t−τ1

s−τ4

By (2.3) and (2.5), we have 

2

Ex(t) − x(t − τ1 ) ≤ 2(τ1 k3 + β) 

t



+2k7 δτ4 Thus



t

0



τ



2

t−τ1

Ex(s) ds + 2τ1 k4



t

t−τ1

s

t

t−τ1  t

Ex(s − τ3 )2 ds + 2k5 k6 τ1 τ2

+2γ t−τ1

t

Ex(s − τ1 )2 ds 

t−τ1

s

Ex(r)2 drds

s−τ2

Ex(r)2 drds.

(2.27)

s−τ4

exp(2εs)Ex(s) − x(s − τ1 )2 ds

0

exp(2εs)Ex(s) − x(s − τ1 )2 ds +



t

exp(2εs)Ex(s) − x(s − τ1 )2 ds  s

 t exp(2εs) Ex(u)2 du ds ≤ 2τ exp(2ετ )(Eξ2 + Exτ 2 ) + 2(τ1 k3 + β)

=

 +2τ1 k4  +2γ

t

t

 exp(2εs)

τ

 exp(2εs)

τ



+2k5 k6 τ1 τ2  +2k7 δτ4

t

t

s

τ

s−τ1

2 Ex(u − τ1 ) du ds

s

s−τ1

Ex(u − τ3 )2 du ds

s−τ1

 exp(2εs)

τ

 exp(2εs)

τ

τ

s

s



u

Ex(r)2 drdu ds

s−τ1 u−τ2  u

s−τ1

Ex(r) drdu ds. 2

(2.28)

u−τ4

By changing the order of integrations, one arrives at the following inequalities:  u+τ1  s



 t  t exp(2εs) Ex(u)2 du ds ≤ Ex(u)2 exp(2εs)ds du τ

s−τ1

0

≤ τ1 exp(2ετ1 )

 0

u

t

exp(2εu)Ex(u)2 du,

(2.29)

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t

 exp(2εs)

τ

 ≤

t

0

s

s−τ1

Ex(u − τ1 )2

Ex(u − τ1 )2 du ds



u+τ1

u

 t exp(2εs)ds du ≤ τ1 exp(2ετ1 ) exp(2εu)Ex(u − τ1 )2 du

≤ τ12 exp(4ετ1 )Eξ2 + τ1 exp(4ετ1 ) 

t

 exp(2εs)

τ

 ≤

t

0

s

s−τ1

Ex(u − τ3 )

2



t

0

exp(2εu)Ex(u)2 du,

t

τ

≤ τ2

 exp(2εs)

Ex(u − τ3 )2 du ds



u+τ1

u





s

s−τ1 s



t

 t exp(2εs)ds du ≤ τ1 exp(2ετ1 ) exp(2εu)Ex(u − τ3 )2 du 

Ex(r) drdu ds

u

t

s−τ1 −τ2

0

exp(2εu)Ex(u)2 du,

(2.31)

0

2

u−τ2

exp(2εs) τ

(2.30)

0

≤ τ1 τ3 exp(2ε(τ1 + τ3 ))Eξ2 + τ1 exp(2ε(τ1 + τ3 )) 

Vol.31 Ser.B

Ex(r)2 drds ≤ τ2 (τ1 + τ2 )



 t 2 2 ≤ τ2 (τ1 + τ2 ) τ Eξ + exp(2εs)Ex(s) ds ,



t

exp(2εs)Ex(s)2 ds

τ −τ1 −τ2

(2.32)

0

and



 exp(2εs)

t

τ



≤ τ4

s

s−τ1 s



t



2

u−τ4

exp(2εs) s−τ1 −τ4

τ

Ex(r) drdu ds

u

Ex(r)2 drds ≤ τ4 (τ1 + τ4 )



 t 2 2 ≤ τ4 τ1 + τ4 )(τ Eξ + exp(2εs)Ex(s) ds .



t

exp(2εs)Ex(s)2 ds

τ −τ1 −τ4

(2.33)

0

Substitution of (2.29)–(2.33) into (2.28) gives  t exp(2εs)Ex(s) − x(s − τ1 )2 ds 0

≤ M2 + 2[(τ1 k3 + β)τ1 exp(2ετ1 ) + τ12 k4 exp(4ετ1 ) + γτ1 exp(2ε(τ1 + τ3 ))  t +k5 k6 τ1 τ22 (τ1 + τ2 ) + k7 δτ42 (τ1 + τ4 )] exp(2εu)Ex(u)2 du,

(2.34)

0

where M2 : = 2τ exp(2ετ )(Eξ2 + Exτ 2 ) + 2Eξ2 [τ13 k4 exp(4ετ1 ) +γτ1 τ3 exp(2ε(τ1 + τ3 )) + k5 k6 τ τ1 τ22 (τ1 + τ2 ) + k7 δτ τ42 (τ1 + τ4 )].

(2.35)

Moreover, with the H¨older inequality and the inequality (a + b)2 ≤ 2(a2 + b2 ) (∀a, b ∈ R), we have  s 2  t  t  s     exp(2εs)E  h2 (s, r, x(r))dr  ds ≤ τ4 exp(2εs) Eh2 (s, r, x(r))2 drds 0

≤ τ4 k7



s−τ4  s

t

exp(2εs) 0

s−τ4

Ex(r)2 drds ≤ τ4 k7



0 t

−τ4

Ex(r)2

s−τ4 r+τ4



exp(2εs)dsdr r

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Y.L. Niu et al: A DELAY-DEPENDENT STABILITY CRITERION

 ≤ τ4 k7 exp(2ετ4 )

t

exp(2εr)Ex(r)2 dr

−τ4

 ≤ τ4 k7 exp(2ετ4 )

1819

0

exp(2εr)Ex(r)2 dr +

−τ4

≤ τ42 k7 exp(2ετ4 )Eξ2 + τ4 k7 exp(2ετ4 )





t

exp(2εr)x(r)2 dr



0 t

exp(2εu)Ex(u)2 du,

(2.36)

0

and 2  h1 (s, r, x(r))dr   ds 0 s−τ2  s 2  t  t    ds exp(2εs)Ex(s)2 ds + 2 exp(2εs)E  h (s, r, x(r))dr ≤2 1   0 0 s−τ2  t  t ≤2 exp(2εs)Ex(s)2 ds + 2τ2 k6 exp(2ετ2 ) exp(2εs)Ex(s)2 ds 

t

    exp(2εs)E x(s) −

s

0

0

+2τ22 k6 exp(2ετ2 )Eξ2 .

(2.37)

Substituting (2.25), (2.34), (2.36) and (2.37) into (2.24) gives  t exp(2εs)Ex(s)2 ds, t ≥ τ, exp(εt)EV (t, x(t)) ≤ M3 + ρ(ε, τ1 , τ2 , τ3 , τ4 )

(2.38)

0

where M3 := M1 + [γτ3 exp(2ετ3 )+ δτ42 k7 exp(2ετ4 )+ 2θ−1 k2 k6 τ22 exp(2ετ2 )]Eξ2 + θ−1 k1 M2 , (2.39) and ρ(ε, τ1 , τ2 , τ3 , τ4 ) = (α + β + θ + 2ε) + γ exp(2ετ3 ) + δτ4 k7 exp(2ετ4 ) + 2θ−1 k1 [(τ1 k3 + β)τ1 × exp(2ετ1 ) + τ12 k4 exp(4ετ1 ) + γτ1 exp(2ε(τ1 + τ3 )) + k5 k6 τ1 τ22 (τ1 + τ2 ) +k7 δτ42 (τ1 + τ4 )] + 2θ−1 k2 (1 + τ2 k6 exp(2ετ2 )).

(2.40)

Take ε = 0. Then ρ(0, τ1 , τ2 , τ3 , τ4 ) = (α + β + θ) + γ + δτ4 k7 + 2θ−1 k1 [(τ1 k3 + β)τ1 + τ12 k4 + γτ1 +k5 k6 τ1 τ22 (τ1 + τ2 ) + k7 δτ42 (τ1 + τ4 )] + 2θ−1 k2 (1 + τ2 k6 ) ≤ ρ(0, τ˜),

(2.41)

where τ k3 + β)˜ τ + τ˜2 k4 + γ τ˜ ρ(0, τ˜) = (α + β + θ) + γ + δ˜ τ k7 + 2θ−1 k1 [(˜ +2k5 k6 τ˜4 + 2k7 δ˜ τ 3 ] + 2θ−1 k2 (1 + τ˜k6 ) = 4k1 k5 k6 θ−1 τ˜4 + 4k1 k7 δθ−1 τ˜3 + 2k1 (k3 + k4 )θ−1 τ˜2 + (2(k1 (β + γ) +k2 k6 )θ−1 + δk7 )˜ τ + α + β + γ + θ + 2θ−1 k2 .

(2.42)

Invoking (2.14) and (2.15), ρ(0, σ) = 0.

(2.43)

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From the known condition τ˜ < σ and the mono-increasing property of function ρ(0, ·), it can be concluded that ρ(0, τ˜) < 0. (2.44) A combination of (2.41) and (2.44) yields ρ(0, τ1 , τ2 , τ3 , τ4 ) ≤ ρ(0, τ˜) < 0.

(2.45)

Hence one can find an ε > 0 small enough for ρ(ε, τ1 , τ2 , τ3 , τ4 ) ≤ 0.

(2.46)

Then, by (2.38), it holds exp(εt)EV (t, x(t)) ≤ M3 ,

t ≥ τ.

(2.47)

Moreover, by the definition of V (t, x(t)) we find that EV (t, x(t)) ≥ Ex(t)2 .

(2.48)

This, together with (2.47), yields Ex(t)2 ≤ M3 exp(−εt),

t ≥ τ.

(2.49)

1 log(Ex(t)2 ) ≤ −ε < 0, t

(2.50)

Therefore, lim sup

t→∞

which proves the theorem. When taking θ = 1, one obtains easily from Theorem 1 the following consequence. Corollary 1 Assume that SDIDE (2.1) satisfies conditions (2.2)–(2.7) and α + β + γ + 1 + 2k2 < 0.

(2.51)

Then (2.1) is exponentially stable in mean-square whenever τ˜ < σ, with σ > 0 being the unique positive root of the equation C(σ) = 0, (2.52) where C(σ) = 4k1 k5 k6 σ 4 + 4k1 k7 δσ 3 + 2k1 (k3 + k4 )σ 2 + (2(k1 (β + γ) + k2 k6 ) + δk7 )σ +α + β + γ + 1 + 2k2 .

(2.53)

Remark 1 If there are no integral terms, (2.1) is reduced to a stochastic delay differential equation. Then the solution of (2.14) is  σ = 2(k3 + k4 )( (β + γ)2 − 2θ(α + β + θ + δ)(k3 + k4 )/k1 − β − γ). (2.54) This condition is in accordance with the result without Markovian switching in [13]. Consider the one-dimensional linear stochastic system ⎧  t ⎪ ⎪ dx(t) = [ax(t) + bx(t − τ ) + c x(s)ds]dt ⎪ 1 ⎪ ⎪ ⎪ t−τ2 ⎨  t ˜ x(s)ds]dw(t), t ≥ 0, +[˜ ax(t) + bx(t − τ3 ) + c˜ ⎪ ⎪ ⎪ t−τ4 ⎪ ⎪ ⎪ ⎩ x(t) = ψ(t), t ∈ [−τ, 0].

(2.55)

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One can easily check that conditions (2.2)–(2.7) are satisfied with α = 2(a + b + c), β = 3˜ a2 , γ = 3˜b2 , δ = 3˜ c2 , k1 = 2b2 , k2 = 2c2 , k3 = 3a2 , k4 = 3b2 , k5 = 3c2 , k6 = 1, k7 = 1. An application of Theorem 1 leads to the following results. Corollary 2 Assume that SDIDE (2.55) satisfies 2(a + b + c) + 3(˜ a2 + ˜b2 ) + 4|c| < 0.

(2.56)

Then (2.55) is exponentially stable in mean-square whenever τ˜ < σ, with σ > 0 being the unique positive root of the equation C(σ) = 0. (2.57) Here C(σ) = 24b2 c2 σ 4 + 24b2 c˜2 θ−1 σ 3 + 12b2(a2 + b2 )θ−1 σ 2 + (4(3b2 (˜ a2 + ˜b2 ) + c2 )θ−1 + δ)σ +2(a + b + c) + 3(˜ a2 + ˜b2 ) + θ + 4c2 θ−1 , (2.58) and θ > 0 satisfies 2(a + b + c) + 3(˜ a2 + ˜b2 ) + θ + 4c2 θ−1 < 0. Corollary 3

(2.59)

Assume that SDIDE (2.55) satisfies 2(a + b + c) + 3(˜ a2 + ˜b2 ) + 1 + 4c2 < 0.

(2.60)

Then (2.55) is exponentially stable in mean-square whenever τ˜ < σ, with σ > 0 being the unique positive root of the equation C(σ) = 0. (2.61) Here C(σ) = 24b2 c2 σ 4 + 24b2c˜2 σ 3 + 12b2 (a2 + b2 )σ 2 + (4(3b2 (˜ a2 + ˜b2 ) + c2 ) + δ)σ +2(a + b + c) + 3(˜ a2 + ˜b2 ) + 1 + 4c2 .

(2.62)

Remark 2 Results in Corollary 1 and Corollary 3 depend only on the system parameters. However, one can obtain a larger bound σ for the delay terms τi (i = 1, 2, 4) from (2.15) or (2.58) because of the flexible parameter θ. This will be illustrated in Section 3.

3

Numerical Experiments

In order to give an intuitionistic illustration to the above stability result, in (2.55), we take a = −4, b = 0.2, c = 0.1, ˜ a = 0.1, ˜b = 1, c˜ = 0.2, τ1 = 0.5, τ2 = 0.25, τ3 = 1, τ4 = 0.5 and ψ(t) = t + 1. In view of (2.59), we should choose θ ∈ (0.0092, 4.3608) to make sure that (2.57) has at least a positive root. Substituting these parameters into (2.57) and solving it by MATLAB yield that the best choice is θ = 2.1410 in order to make σ as large as possible. In this case, σ = 0.7334 which implies that the stability condition τ˜ = max {τi } < 0.7334 i=1,2,4

(3.1)

is satisfied. However, σ = 0.6161 from Corollary 3. This shows that for certain choices of the parameter θ, one can obtain a larger bound than θ is restricted to 1. We solve (2.55) on [0, 10]

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ACTA MATHEMATICA SCIENTIA

Vol.31 Ser.B

by the Euler-Maruyama method with step size h = 2−8 . The obtained numerical solution, averaged over 5000 trajectories to indicate the trend of the solution, is plotted in Fig.1. The figure shows that the solution is exponentially stable in mean-square.

Fig.1

The numerical solution for (2.55) with a = −4, b = 0.2, c = 0.1, a ˜ = 0.1, ˜b = 1, c˜ = 0.2,

τ1 = 0.5, τ2 = 0.25, τ3 = 1, τ4 = 0.5 and ψ(t) = t + 1.

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