D(r(t)) = [A(r(t))x(t) + F (x(t), r(t), u) + H(r(t))u]dt + G(x(t), r(t), u)dw(t), {r(t)} t≥0 S = {1, 2, , N} Γ = (γ ij ) N×N (r(t + ∆) = j|r(t) = i) =  γ ij ∆ + o(∆) i = j 1 + γ ii ∆ + o(∆) i = j ∆ > 0 γ ij ≥ 0 i j γ ii = −  i=j γ ij . w(t) = (w 1 t , , w l t ) T l (Ω, F, ) |.| A A T A = sup{|Ax| : |x| = 1} (1.1) A(i), D(i) H(i) R q×q , R q×q R q×p k 1 (i), , k p (i)  p j=1 k j (i) = q A(i) A(i) =       A k 1 (i) 0 ··· 0 0 0 0 ··· 0 A k p (i)       , A k j (i) , 1  j  p R k j ×k J A k j (i) =       0 1 0 ··· 0 0 0 0 1 ··· 0 0 ··· 0 0 0 ··· 0 1 0 0 0 0 ··· 0 0       ; H(i) H(i) =        b k 1 (i) 0 0 ··· 0 0 0 b k 2 (i) 0 0 0 ··· ··· 0 0 0 ··· b k p−1 (i) 0 0 0 0 ··· 0 b k p (i)        , b k j (i) , 1  j  p R k j b k j (i) =      0 0 1      D(i) A(i) F = (F 1 , , F q ) : R q ×S ×R p −→ R q G = (G 1 , , G q ) : R q ×S ×R p −→ R q×l F (0, i, 0) = G(0, i, 0) = 0 λ > 0 ∀j, 1  j  q, x ∈ R q u ∈ R p , |F j (x, i)| + |G j (x, i)|  λ|π j (x)| π j R q R j ; u α ∈ R, α > 1 i ∈ S φ(i) =       α −1 i 0 ··· 0 0 0 0 ··· 0 α −q i       · (A(i), H(i)) K(i) ∈ R p×q (R) M(i) = A(i) + H(i)K(i) P (i) M T (i)P (i)D(i) + D T (i)P (i)M(i) = −I. x ≡ 0 (1) α β Ex(t, t 0 , x 0 ) 2  αx 0  2 e −β(t−t 0 ) , t ≥ t 0 . [3] x ∈ R q u ∈ R p |φ(i)F (x, i, u )|  √ qλ|φ(i)x| |φ(i)G(x, i, u)|  √ qλ|φ(i)x|. [3] α i φ −1 (i)A(i)φ(i) = A(i); α i φ −1 (i)D(i)φ(i) = D(i) K(i) ∈ R p×q K(i) H(i)K(i) = α i φ −1 (i)H(i)K(i)φ(i) K(i) K(i) = α i H T (i)φ −1 (i)H(i)K(i)φ(i), x ∈ R q α −q i |x|  |φ(i)x|  α −1 i |x|. V ∈ C 2,1 (R n × R + × S; R + ) LV R n × R + × S R LV (x, t, i) = V t (x, t, i) + V x (x, t, i)f(x, t, i) + 1 2 trace[g T (x, t, i)V xx (x, t, i)g(x, t, i)] + N  j=1 γ ij V (x, t, i), f(x, t, i) = F (x(t), r(t), u) g(x, t, i) = G(x(t), r(t), u)dw(t) V t = ∂V ∂t , V x = ( ∂V ∂x 1 , , ∂V ∂x n ) V xx = ( ∂ 2 V ∂x i ∂x j ) n×n . [4] V (x, i) ∈ C 2,1 (R n ×R + ×S; R + ) c 1 , c 2 c 3 c 1 |x| 2  V (x, i)  c 2 |x| 2 LV (x, i) < −c 3 |x| 2 (1) u = K(i)x (1) α i , i ∈ S  −α 2 i φ 2 (i) +  N j=1 γ ij D T (j)φ(j)D(j)  (2λα i D √ q + qλ 2 )P φ(i)  (2λα i D √ q + qλ 2 )P φ(i) −I  < 0. [2] R = R T M = M T M + NR −1 N T < 0  M N N T −R  < 0 R = R T i ∈ S K(i) ∈ R p×q (R) A(i) + H(i)K(i) = α i φ −1 (i)M(i)φ(i). V (x, i) = (D(i)x) T φ(i)P (i)φ(i)(D (i)x). LV (x, i) = [G T (x, i, u)φ(i)P (i)φ(i)G(x, i, u)]+ 2x T D T (i)φ(i)P (i)φ(i)F (x, i, u) + x T N  j=1 γ ij D T (j)φ(j)P (j)φ(j)D(j)x + x T [(A(i) + H(i)K(i)) T φ(i)P (i)φ(i)D (i) + D T (i)φ(i)P (i)φ(i)(A(i) + H(i)K(i))]x. P  = max{P (i), i ∈ S} D = max{D(i), i ∈ S} LV (x, i) = 2α i x T φ(i)D T (i)P (i)φ(i)F (x, i, u)+ + x T N  j=1 γ ij D T (j)φ(j)P (j)φ(j)D(j)x + α 2 i x T φ(i)[(M(i) T P (i)D(i) + D T (i)P (i)M(i)]φ(i)x + [G T (x, i, u)φ(i)P (i)φ(i)G(x, i, u)]  −α 2 i x T φ(i)φ(i)x + x T N  j=1 γ ij D T (j)φ(j)P (j)φ(j)D(j)x + P   2α i D|φ(i)x||φ(i)F (x, i, u)| + |φ(i)G(x, i, u)| 2  |φ(i)F (x, i, u )|  √ qλ|φ(i)x| |φ(i)G(x, i, u)| 2  qλ 2 |φ(i)x| 2 . LV (x, i)   −α 2 i +(2α i D √ qλ+qλ 2 )P   |φ(i)x| 2 +x T  N j=1 γ ij D T (j)φ(j)P (j)φ(j)D(j)x. c > 0 LV (x, i)  −c|x| 2 . (1)