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SUFICIENT CONDITION FOR EXPONENTIAL STABILITY FOR A CLASS OF STOCHASTIC DELAY EQUATIONS

Nguyen Thanh Dieu (a)

Abstract. In this article, we study the exponential stability in mean square for

a class of stochastic differential delay equations of the form

This equation is regarded as a stochastically perturbed equation of a nonlinear delay equation with the exponential stability

dx(t) = f (x(t), x(t − τ ), t)dt + σ(t, x(t))dw(t).

This result show that a damped stochastic perturbation can be tolerate by second equation without losing the property of exponential stability.

dx(t) = f (x(t), x(t − τ ), t)dt.

1. Introduction

Stochastic differential equations is used to provide a mathematical model for natu- ral dynamical systems in physical, biological, medical and social sciences. However in many circumstances, the future state depends not only on the present state but also on its history. Stochastic differential equations give a mathematical formulation for such systems. The stability problem for such equations has been investigated by many authors [1-4]. Recently in [4], X. Mao has studied the almost sure exponential stability for a class of differential equations is of form

In this paper we will study the exponential stability in mean square for a class of stochastic defferential delay equations of the form

dx(t) = f (x(t), x(t−τ ), t)dt+σ(t)dw(t). (1.1)

(1.2) dx(t) = f (x(t), x(t−τ ), t)dt+σ(t, x(t))dw(t).

2. Preliminaries

Throughout this paper let (Ω, =, {=t}t>0, P ) be a complete probability space with a filtration {=t}t>0, which is right continous and contains all P- null sets. Denote by |x| the Euclidean norm of a vector x ∈ Rn. Denote by kAk the operator norm of a matrix A, i.e. kAk = sup{|Ax| : |x| = 1}. Also denote by BT the transpose of matrix

1 Nh¸n b(cid:181)i ng(cid:181)y 07/5/2007. S(cid:246)a ch(cid:247)a xong ng(cid:181)y 10/10/2007.

B. For a square matrix A = (aij), Trace(A) = P aii. Let τ be a positive constant and by C([−τ, 0]; Rd) denote the family of all continuous Rd− valued functions defined on [−τ, 0]. By L2 ([−τ, 0]; Rd) denote the family of =t− measurable, C([−τ, 0]; Rd)-valued =t random variables ξ = {ξ(u) : −τ 6 u 6 0} such that

E = sup

−τ 6u60

Consider stochastic differential equations of the form

kξk2 E|ξ(u)|2 < ∞.

dx(t) = f (x(t), x(t−τ ), t)dt+σ(t, x(t))dw(t); on t ≥ 0 (2.1)

with initial data x(t) = ξ(u) on −τ 6 u 6 0; where f : Rd × Rd × R+ → Rd, σ : Rd×R+ → Rd×m and w is an m- dimensional Brownian motion and ξ ∈ L2 =0 Assume the equation has a unique solution that is denoted by x(t, ξ).

([−τ, 0]; Rd).

Definition 2.1. The stochastic differential equations (2.1) is said to be exponential stable in mean square if there is a pair of positive constant δ and K such that for any initial data ξ ∈ L2 =0

([−τ, 0]; Rd)

Ee−δt, ∀t ≥ 0.

We refer to δ as the rate constant and K as the growth constant.

Lemma 2.2. (Gronwall- Bellman lemma [2]) Let u(t) and v(t) be continuous noneg-

ative functions and let N0 be a positive constant such that for t ≥ s

E|x(t, ξ)| 6 Kkξk2 (2.2)

s

Then for t ≥ s

Z t u(t) 6 N0 + u(t1)v(t1)dt1.

s

Z t (2.3) v(t1)dt1}. u(t) 6 N0exp{

3. Main results

Theorem 3.1. Let c1 − c3 be positive constants. Asume (i) 2xT f (x, y, t) 6 −c1|x|2 + c2|y|2, (ii) T race(σ(t, x)σT (t, x)) 6 c3|x|2, (iii) c2ec1τ + c3 < c1,

for all x, y ∈ Rd; stability in mean square.

t > 0. Then the stochastic differential equations (2.1) is exponential

([−τ, 0]; Rd) Fix ξ arbitrarily and write x(t, ξ) = x(t) simple. By Ito’s

Proof. For all ξ ∈ L2 F0 formula and assumption,

ec1t|x(t)|2 = |x(0)|2 + M (t) + N (t)

0

for all t ≥ 0, where Z t ec1sxT (s)σ(s, x(s))dw(s). M (t) = 2

0

Z t N (t) = ec1s(c1|x(s)|2 + 2x(s)T f (x(s), x(s − τ ), s) + trace(σ(s, x(s))σT (s, x(s)))ds.

By (i), (ii) we have

0

Z t ec1t|x(t)|2 6 |x(0)|2+M (t)+ (4.4) ec1s(c2|x(s−τ )|2+c3|x(s)|2)ds.

0

0

τ

But Z t Z τ Z max {τ,t} ec1s|x(s − τ )|2ds 6 ec1s|x(s − τ )|2ds + ec1s|x(s − τ )|2ds

0

0

Z τ Z t 6 ec1s|x(s−τ )|2ds+ec1τ ec1s|x(s)|2ds. (4.5).

From inequalities (4.5) and (4.4) follow that

0

0

Z τ Z t ec1t|x(t)|2 6 |x(0)|2 + M (t) + ec1sc2|x(s − τ )|2ds + (c3 + c2ec1τ )ec1s|x(s)|2ds

E; E|x(0)|2 6 kξk2 E

0

because EM (t) = 0, moreover we have Z τ (ec1τ − 1)kξk2 ec1sc2E|x(s − τ )|2ds 6 c2 c1

0

0

so that Z t Z τ ec1tE|x(t)|2 6 E|x(0)|2 + (c3 + c2ec1τ )ec1sE|x(s)|2ds ec1sc2E|x(s − τ )|2ds +

E+

0

Z t (4.6). (ec1τ −1))kξk2 6 (1+ (c3+c2ec1τ )ec1sE|x(s)|2ds. c2 c1 From (4.6) and applying lemma 2.2 with

u(t) = ec1tE|x(t)|2; v(t) = c3 + c2ec1τ ; N0 = (1 + (ec1τ − 1))kξk2 E c2 c1

we have

Ee(c3+c2ec1τ )t.

ec1tE|x(t)|2 6 (1 + (ec1τ − 1))kξk2 c2 c1 Hence we obtain

Ee(c3+c2ec1τ −c1)t

(ec1τ − 1))kξk2 E|x(t)|2 6 (1 + c2 c1

By assumptions (iii) we can rewrite

Ee−δt,

E|x(t)|2 6 Kkξk2

(ec1τ − 1) > 0 and δ = c1 − c3 − c2ec1τ > 0.

where K = 1 + c2 c1 In other words,the stochastic differential equations (2.1) is exponential stability in mean (cid:3) square. The proof is completed.

Acknowledgement. The author expresses his gratefulness to Professor Phan Duc Thanh for his suggestions.

References

[1] L. Arnold, Stochastic differential equation: Theory and Applications, New York, Springer, 1970. [2] R. Z. Hasminski, Stochastic stability of differential equations, Sythoff and Noard- hoff, Alphen aan den Rijn, The Netherlands Rockville, Maryland, USA, 1980. [3] X. Mao, Exponential stability for stochastic differential delay equations in Hilbert space, Q. J. Math, Oxford, Vol 42, 1991, pp. 77-85. [4] X. Mao, Almost sure exponential stability of delay equations with damped stochastic perturbation, Stochastic analysis and application, Vol 19, 2001, pp. 67-84.

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cæa ph›‹ng tr(cid:215)nh vi ph'n ng(cid:201)u nhi“n cª tr(cid:212) d„ng

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dx(t) = f (x(t), x(t − τ ), t)dt + σ(t, x(t))dw(t).

K(cid:213)t qu¶ n(cid:181)y ch(cid:216) ra r»ng vi(cid:214)c l(cid:181)m nhi(cid:212)u n(cid:181)y kh«ng l(cid:181)m m˚t t(cid:221)nh (cid:230)n fi(cid:222)nh cæa nª.

(a) Khoa To‚n, tr›Œng §„i h(cid:228)c Vinh.

dx(t) = f (x(t), x(t − τ ), t)dt.