Báo cáo nghiên cứu khoa học: "Về một điều kiện đủ cho tính ổn định mũ của một lớp phương trình vi phân ngẫu nhiên có trễ"

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Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của trường đại học vinh năm 2008 tác giả: 4. Nguyễn Thanh Diệu, Về một điều kiện đủ cho tính ổn định mũ của một lớp phương trình vi phân ngẫu nhiên có trễ ..Khoa học (trong tiếng Latin scientia, có nghĩa là "kiến thức" hoặc "hiểu biết") là các nỗ lực thực hiện phát minh, và tăng lượng tri thức hiểu biết của con người về cách thức hoạt động của thế giới vật chất xung quanh. Thông qua các phương pháp kiểm soát, nhà khoa học sử...

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Nội dung Text: Báo cáo nghiên cứu khoa học: "Về một điều kiện đủ cho tính ổn định mũ của một lớp phương trình vi phân ngẫu nhiên có trễ"

SUFICIENT CONDITION FOR EXPONENTIAL STABILITY FOR A CLASS OF STOCHASTIC DELAY EQUATIONS (a) Nguyen Thanh Dieu 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 (Ω, , { t }t 0 , P ) Throughout this paper let be a complete probability space with a { t }t 0 , filtration which is right continous and contains all P- null sets. Denote by Rn . |x| x∈ A the Euclidean norm of a vector Denote by the operator norm of a BT A = sup{|Ax| : |x| = 1}. matrix A, i.e. Also denote by the transpose of matrix 1 NhËn bµi ngµy 07/5/2007. Söa ch÷a xong ngµy 10/10/2007.
A = (aij ), Trace(A) = aii . τ B. For a square matrix Let be a positive constant and C ([−τ, 0]; Rd ) Rd − by denote the family of all continuous valued functions defined on L2 ([−τ, 0]; Rd ) C ([−τ, 0]; Rd )-valued [−τ, 0]. t− By denote the family of measurable, t ξ = {ξ (u) : −τ 0} such that u random variables 2 sup E |ξ (u)|2 < ∞. ξ = E −τ u 0 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) x(t) = ξ (u) on −τ u 0; where f : Rd × Rd × R+ → Rd , with initial data σ : Rd ×R+ → Rd×m and 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 2 ([−τ, 0]; Rd ) initial data ξ ∈ L 0 2 −δt E |x(t, ξ )| ∀t ≥ 0. Kξ Ee , (2.2) δ K We refer to as the rate constant and as the growth constant. u(t) and v (t) be continuous noneg- Lemma 2.2. (Gronwall- Bellman lemma [2]) Let ative functions and let N0 be a positive constant such that for t ≥ s t u(t) N0 + u(t1 )v (t1 )dt1 . s t≥s Then for t N0 exp{ v (t1 )dt1 }. u(t) (2.3) s 3. Main results c1 − c3 be positive constants. Asume Theorem 3.1. Let T −c1 |x|2 + c2 |y |2 , (i) 2x f (x, y, t) T c3 |x|2 , (ii) T race(σ (t, x)σ (t, x)) cτ (iii) c2 e 1 + c3 < c1 , d for all x, y ∈ R ; t 0. Then the stochastic differential equations (2.1) is exponential stability in mean square.
Proof. For all ξ ∈ L2 0 ([−τ, 0]; Rd ) Fix ξ arbitrarily and write x(t, ξ ) = x(t) simple. By Ito’s F formula and assumption, ec1 t |x(t)|2 = |x(0)|2 + M (t) + N (t) for all t ≥ 0, where t ec1 s xT (s)σ (s, x(s))dw(s). M (t) = 2 0 t ec1 s (c1 |x(s)|2 + 2x(s)T f (x(s), x(s − τ ), s) + trace(σ (s, x(s))σ T (s, x(s)))ds. N (t) = 0 By (i), (ii) we have t ec1 t |x(t)|2 |x(0)|2 + M (t)+ ec1 s (c2 |x(s − τ )|2 + c3 |x(s)|2 )ds. (4.4) 0 But max {τ,t} t τ ec1 s |x(s − τ )|2 ds ec1 s |x(s − τ )|2 ds + ec1 s |x(s − τ )|2 ds 0 0 τ τ t ec1 s |x(s−τ )|2 ds+ec1 τ ec1 s |x(s)|2 ds. (4.5). 0 0 From inequalities (4.5) and (4.4) follow that τ t ec1 t |x(t)|2 |x(0)|2 + M (t) + ec1 s c2 |x(s − τ )|2 ds + (c3 + c2 ec1 τ )ec1 s |x(s)|2 ds 0 0 because EM (t) = 0, moreover we have τ c2 c1 τ ec1 s c2 E |x(s − τ )|2 ds 2 2 2 (e − 1) ξ E ; E |x(0)| ξ E c1 0 so that τ t ec1 t E |x(t)|2 E |x(0)|2 + ec1 s c2 E |x(s − τ )|2 ds + (c3 + c2 ec1 τ )ec1 s E |x(s)|2 ds 0 0 t c2 c1 τ (e −1)) ξ 2 + (c3 +c2 ec1 τ )ec1 s E |x(s)|2 ds. (1+ (4.6). E c1 0 From (4.6) and applying lemma 2.2 with c2 c1 τ u(t) = ec1 t E |x(t)|2 ; v (t) = c3 + c2 ec1 τ ; N0 = (1 + 2 (e − 1)) ξ E c1 we have c2 c1 τ 2 (c3 +c2 ec1 τ )t ec1 t E |x(t)|2 (e − 1)) ξ (1 + Ee . c1 Hence we obtain c2 c1 τ 2 (c3 +c2 ec1 τ −c1 )t E |x(t)|2 (e − 1)) ξ (1 + Ee c1
By assumptions (iii) we can rewrite 2 −δt E |x(t)|2 Kξ Ee , where K = 1 + c2 (ec1 τ − 1) > 0 and δ = c1 − c3 − c2 ec1 τ > 0. c1 In other words,the stochastic diﬀerential equations (2.1) is exponential stability in mean 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. tãm t¾t VÒ mét ®iÒu kiÖn ®ñ cho tÝnh æn ®Þnh mò cña mét líp ph¬ng tr×nh vi ph©n ngÉu nhiªn cã trÔ Trong bµi b¸o nµy,chóng t«i nghiªn cøu tÝnh æn ®Þnh mò b×nh ph¬ng trung b×nh cña ph¬ng tr×nh vi ph©n ngÉu nhiªn cã trÔ d¹ng dx(t) = f (x(t), x(t − τ ), t)dt + σ (t, x(t))dw(t). Ph¬ng tr×nh nµy ®îc xem nh lµ ph¬ng tr×nh ngÉu nhiªn cña phi tuyÕn æn ®Þnh mò cã trÔ dx(t) = f (x(t), x(t − τ ), t)dt. KÕt qu¶ nµy chØ ra r»ng viÖc lµm nhiÔu nµy kh«ng lµm mÊt tÝnh æn ®Þnh cña nã. (a) Khoa To¸n, trêng §¹i häc Vinh.