Báo cáo nghiên cứu khoa học: "Về tính ổn định tiệm cận bình ph­ương trung bình của hệ ph­ương trình vi phân ngẫu nhiên có trễ" ppt

 dx(t) = (Ax(t) +  r i=1 A i x(t − h i ))dt + Bx(t)dW (t) x(u) = ξ(u), ∀u ∈ [−h, 0]; t ≥ 0 . (2.1) A, A i , B ∈ R d×d , W (t) h i ∈ [0, h]. (Ω, , { t } t0 , P ) { t } t0 x x ∈ R n A A A = sup{Ax : x = 1} B T B P = (p ij ) n×n λ max (P ); λ min (P ) P h C([−h, 0]; R n ) [−h, 0] R n L 2  t ( [−h, 0]; R n )  t C([−h, 0]; R n )− ξ = {ξ(u) : −h  u  0} ξ 2 E = sup −hu0 Eξ(u) 2 < ∞, E(.) dx(t) = (Ax(t) + A 1 x(t − h))dt + Bx(t)dw(t) t  0 (3.1) x(u) = ξ(u) −h  u  0, A, A 1 , B ∈ R n×n w (Ω, , { t } t0 , P ) ξ ∈ L 2  0 ( [ −h, 0 ] ; R n ) . (3.1) x(u) = ξ(u) −h  u  0 x(t, ξ) (3.1) ε > 0 r > 0 E|x(t, ξ)| 2 < ε t > t 0 ; ξ ∈ L 2  0 ([−h, 0]; R n ) ξ  r. (3.1) E|x(t, ξ)| 2 −→ 0 t −→ ∞. (3.2)  dx(t) = Ax(t)dt + Bx(t)dW (t) x(t 0 ) = I; t  t 0 , (3.3) A, B n × n W (t) (Ω, , { t } t0 , P ) Φ(t, s) (3.3) (3.3) K > 0; δ > 0 EΦ(t, t 0 ) 2  Ke −δ(t−t 0 ) . Q A T Q + QA + B T QB + P = 0. (3.4) (3.4) Q = E  ∞ 0 Φ T (t, 0)P Φ(t, 0)dt (3.5) Φ(t, 0) P [6] S ∈ R n×n P, Q ∈ R n×n (P x, x) + 2( Qy, x) − (Sy, y)  ((P + QS −1 Q T )x, x). (3.6) & [5], p.169 x(t, ξ) x t (ξ) = {x(t + θ, ξ) : −h  θ  0} V : R + × C([−h; 0]; R n ) −→ R c 1 |ϕ(0)| 2  V (t, ϕ)  c 2 ϕ 2 ∀(t, ϕ) ∈ R + × C([−h, 0]; R n ) EV (t 2 , x t 2 (ξ)) − EV (t 1 , x t 1 (ξ))  −c 3  t 2 t 1 E|x(s, ξ)| 2 ds t 2 > t 1  0, c 1 ; c 2 ; c 3 Q x T Qx  λ max (P )K δ x 2 . (4.1) (3.3) (Qx, x) = E  ∞ 0 (P Φ(t, 0)x, Φ(t, 0)x)dt  λ max (P )x 2 E  ∞ 0 Φ(t, 0) 2 dt  λ max (P )x 2 K  ∞ 0 e −δt dt  λ max (P )K δ x 2 , (Qx, x)  λ max (P )K δ x 2 . A 1  < δ 2K . (4.2) x(t) (3.1) x t (s) = x(t + s), s ∈ [−h; 0] Q (3.4) P = αI. V (x t , t) = (Qx(t), x(t)) +  0 −h x(t + τ) 2 dτ. (4.1) λ min (Q)x(t) 2  V (x t , t)  ( λ max (P )K δ + h)x t  2 . (4.3) dV (x t , t) = {((A T Q + QA + B T QB)x(t), x(t)) + 2(QA 1 x(t), x(t − h)) +x(t) 2 − x(t − h) 2 }dt + M(t). M(t) = 2x(t) T QBx(t)dW (t). (3.6) (4.1) 2(QA 1 x(t), x(t − h)) − x(t − h) 2  Q 2 A 1  2 x(t) 2  α 2 K 2 δ 2 A 1  2 x(t) 2 . dV (x t , t)  (−α + 1 + α 2 K 2 δ 2 A 1  2 )x(t) 2 dt + M(t). EM(t) = 0 EV (x t , t) − EV (x 0 , 0)   t 0 (−α + 1 + α 2 K 2 δ 2 A 1  2 )Ex(s) 2 ds. (4.4) (α−1) α 2 α = 2 (α − 1)δ 2 α 2 K 2  δ 2 4K 2 . (4.5) (4.2) (4.5) α 0 2 A 1  2 < (α 0 − 1)δ 2 K 2 α 2 0 c 3 = α 0 − 1 − α 2 0 K 2 δ 2 A 1  2 > 0. (4.6) (4.3), (4.4) (4.6) c 1 = λ min (Q); c 2 = λ max (P )K δ + h; c 3 = α 0 − 1 − α 2 0 K 2 δ 2 A 1  2 r  i=1 A i  2 < δ 2 4rK 2 . (4.7) x(t) (2.1) x t (s) = x(t + s) s ∈ [−h; 0] Q (3.3) P = αI. V (x t , t) = (Qx(t), x(t)) + r  i=1  0 −h i x t (τ) 2 dτ. (4.1) λ min (Q)x(t) 2  V (x t , t)  ( λ max (P )K δ + r  i−1 h i )x t  2 . (4.8) dV (x t , t) = {((A T Q + QA + B T QB)x(t), x(t)) + 2 r  i=1 (QA i x(t), x(t − h i ))+ +rx(t) 2 − r  i=1 x(t − h i ) 2 }dt + M(t). M(t) = 2x T (t)QBx(t)dW (t). (3.6) (4.1) 2(QA i x(t), x(t − h)) − x(t − h i ) 2  Q 2 A i  2 x(t) 2  α 2 K 2 δ 2 A i  2 x(t) 2 . dV (x t , t)  (−α + r + α 2 K 2 δ 2 ( r  i=1 A i  2 ))x(t) 2 .dt + M(t). EM(t) = 0 EV (x t , t) − EV (x 0 , 0)   t 0 (−α + r + α 2 K 2 δ 2 ( r  i=1 A i  2 ))Ex(s) 2 ds. (4.9) (α−r) α 2 α = 2r (α − r)δ 2 α 2 K 2  δ 2 4rK 2 . (4.10) (4.7) (4.10) α 0 r  i=1 A i  2 < (α 0 − r)δ 2 α 2 0 K 2 c 3 = α 0 − r − α 2 0 K 2 δ 2 ( r  i=1 A i  2 ) > 0. (4.11) (4.8), (4.9) (4.11) c 1 = λ min (Q); c 2 = λ max (P )K δ +  r i=1 h i ; c 3 = α 0 − r − α 2 0 K 2 δ 2 (  r i=1 A i  2 )