SUFICIENT CONDITION FOR EXPONENTIAL STABILITYFOR A CLASS OF STOCHASTIC DELAY EQUATIONS Nguyen Thanh Dieua Abstract.. This result show that a damped stochastic perturbation can be tolerat
Trang 1SUFICIENT 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
dx(t) = f (x(t), x(t − τ ), t)dt + σ(t, x(t))dw(t)
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
This result show that a damped stochastic perturbation can be tolerate by second equation without losing the property of exponential stability.
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
dx(t) = f (x(t), x(t−τ ), t)dt+σ(t)dw(t) (1.1)
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, x(t))dw(t) (1.2)
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 NhËn bµi ngµy 07/5/2007 Söa ch÷a xong ngµy 10/10/2007.
Trang 2B 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
= t([−τ, 0]; Rd)denote the family of =t−measurable, C([−τ, 0]; Rd)-valued random variables ξ = {ξ(u) : −τ 6 u 6 0} such that
kξk2E = sup
−τ 6u60
E|ξ(u)|2 < ∞
Consider stochastic differential equations of the form
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×mand w is an m- dimensional Brownian motion and ξ ∈ L2
= 0([−τ, 0]; Rd) Assume the equation has a unique solution that is denoted by x(t, ξ)
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) E|x(t, ξ)| 6 Kkξk2Ee−δt, ∀t ≥ 0 (2.2)
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
u(t) 6 N0+
Z t
s
u(t1)v(t1)dt1 Then for t ≥ s
u(t) 6 N0exp{
Z t
s
3 Main results Theorem 3.1 Let c1− c3 be positive constants Asume
(i) 2xTf (x, y, t) 6 −c1|x|2+ c2|y|2,
(ii) T race(σ(t, x)σT(t, x)) 6 c3|x|2,
(iii) c2ec1 τ+ c3 < c1,
for all x, y ∈ Rd; t > 0.Then the stochastic differential equations (2.1) is exponential stability in mean square
Trang 3Proof For all ξ ∈ L2F0([−τ, 0]; Rd) Fix ξ arbitrarily and write x(t, ξ) = x(t) simple By Ito’s formula and assumption,
ec1 t|x(t)|2 = |x(0)|2+ M (t) + N (t) for all t ≥ 0, where
M (t) = 2
Z t
0
ec1 sxT(s)σ(s, x(s))dw(s)
N (t) =
Z t
0
ec1 s(c1|x(s)|2+ 2x(s)Tf (x(s), x(s − τ ), s) + trace(σ(s, x(s))σT(s, x(s)))ds
By (i), (ii) we have
ec1 t|x(t)|26 |x(0)|2+M (t)+
Z t
0
ec1 s(c2|x(s−τ )|2+c3|x(s)|2)ds (4.4) But
Z t
0
ec1 s|x(s − τ )|2ds 6
Z τ
0
ec1 s|x(s − τ )|2ds +
Z max {τ,t}
τ
ec1 s|x(s − τ )|2ds
6
Z τ
0
ec1 s|x(s−τ )|2ds+ec1 τ
Z t
0
From inequalities (4.5) and (4.4) follow that
ec1 t|x(t)|2 6 |x(0)|2+ M (t) +
Z τ
0
ec1 sc2|x(s − τ )|2ds +
Z t
0
(c3+ c2ec1 τ)ec1 s|x(s)|2ds because EM (t) = 0, moreover we have
Z τ
0
ec1 sc2E|x(s − τ )|2ds 6 c2
c1(e
c 1 τ− 1)kξk2E; E|x(0)|2 6 kξk2E
so that
ec1 tE|x(t)|2 6 E|x(0)|2+
Z τ
0
ec1 sc2E|x(s − τ )|2ds +
Z t
0
(c3+ c2ec1 τ)ec1 sE|x(s)|2ds
6 (1+c2
c1(e
c 1 τ−1))kξk2E+
Z t
0
(c3+c2ec1 τ)ec1 sE|x(s)|2ds (4.6) From (4.6) and applying lemma 2.2 with
u(t) = ec1 tE|x(t)|2; v(t) = c3+ c2ec1 τ; N0= (1 + c2
c1(e
c 1 τ− 1))kξk2E
we have
ec1 tE|x(t)|26 (1 + c2
c1
(ec1 τ− 1))kξk2
Ee(c3 +c 2 e c1τ)t
Hence we obtain
E|x(t)|2 6 (1 +c2
c1
(ec1 τ− 1))kξk2
Ee(c3 +c 2 e c1τ −c 1 )t
Trang 4By assumptions (iii) we can rewrite
E|x(t)|2 6 Kkξk2Ee−δt, where K = 1 +c2
c 1(ec1 τ − 1) > 0 and δ = c1− c3− c2ec1 τ > 0
In other words,the stochastic differential equations (2.1) is exponential stability in mean
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
tóm tắt
Về một điều kiện đủ cho tính ổn định mũ của một lớp phương
trình vi phân ngẫu nhiên có trễ Trong bài báo này,chúng tôi nghiên cứu tính ổn định mũ bình phương trung bình của phương trình vi phân ngẫu nhiên có trễ dạng
dx(t) = f (x(t), x(t − τ ), t)dt + σ(t, x(t))dw(t)
Phương trình này được xem như là phương trình ngẫu nhiên của phi tuyến ổn định
mũ có trễ
dx(t) = f (x(t), x(t − τ ), t)dt
Kết quả này chỉ ra rằng việc làm nhiễu này không làm mất tính ổn định của nó
(a) Khoa Toán, trường Đại học Vinh.