Exponential Stability of Impulsive Stochastic Functional Differential Systems with Delayed Impulses

and Applied Analysis 3 (S2) (h 0 , h)-globally exponentially stable, if there exist constants α > 0 and C ⩾ 1 such that, for all ξ ∈ PC b F t0 ([−τ, 0];Rn) and t 0 ∈ R + , Eh (t, x (t; t 0 , ξ)) ⩽ CEh 0 (t 0 , ξ) e−α(t−t0), t ⩾ t 0 . (6) Remark 4. The (h 0 , h)-stability notions are considered here in the spirit of the work by Lakshmikantham and Liu [26] to unify different stability concepts found in the literature such as the stability of the trivial solution, the partial stability, the conditional stability, and the stability of invariant sets, which would otherwise be treated separately. For example, it is easy to see that (S2) in Definition 3 gives (1) the pth-moment exponential stability of the trivial solution x(t) ≡ 0, if h(t, x) = h0(t, x) = |x|. When p = 2, it is usually called mean square exponential stability; (2) the pth-moment exponentially partial stability of the trivial solution, if h(t, x) = |(x 1 , . . . , x s )|p, 1 ⩽ s ⩽ n and h0(t, x) = |x|; (3) the global exponential stability of the prescribed motion y(t), if h(t, x) = h0(t, x) = |x − y|; (4) the global exponential stability of the invariant setA ∈ R, if h(t, x) = h0(t, x) = d(x, A), where d(x, A) is the distance of x from the set A; (5) the global exponential orbital stability of a periodic solution, if h(t, x) = h0(t, x) = d(x, C), where C is the closed orbit in the phase space. 3. Stability Results In this section, we will develop Lyapunov-Razumikhinmethods and establish some criteria which provide sufficient conditions for the (h 0 , h)-exponential stability and (h 0 , h)uniform stability of ISFDSs-DI. Our results illustrate that the impulses play a positive role in making the continuous flow stable. Theorem 5. Assume that there exist functions V ∈ V 0 , h0, h ∈ Γ and constants c 1 > 0, c 2 > 0, γ > q > 1 such that (i) c 1 h(t, x) ⩽ V(t, x) ⩽ c 2 h0(t, x) for any (t, x) ∈ [t 0 − τ,∞) ×R; (ii) ELV(t, φ) ⩽ b(t)c(EV(t, φ(0))) for all t ∈ [t k−1 , t k ), k ∈ N and those φ ∈ PCF t ([−τ, 0];Rn) satisfying EV (t + θ, φ (θ)) ⩽ γEV (t, φ (0)) on − τ ⩽ θ ⩽ 0, (7) where b, c : R + → R + are continuous; (iii) EV(t k , I k (t k , φ)) ⩽ (1/γ)(1 + β k )sup s∈[−τ,0] EV(t− k + s, φ(s)), where β k ⩾ 0 and ∑∞ k=1 β k < ∞; (iv) ln q > M 1 M 2 , where M 1 = sup t⩾0 ∫ t+μ t b(s)ds, M 2 = sup s>0 {c(s)/s}, and μ = sup k∈N{tk − tk−1}. Then, system (1) is (h 0 , h)-exponentially stable for any time delay τ ∈ (0,∞), and the convergence rate is not greater than min{ln(γ/q)/τ, (ln q −M 1 M 2 )/μ}. Proof. Fix any initial data ξ ∈ PCb F t0 ([−τ, 0);R), and write x(t; t 0 , ξ) = x(t) simply. Let β > 1 be an arbitrary constant, and define α = min{ln(γ/q)/τ, (ln q −M 1 M 2 )/βμ}. We claim that EV (t, x (t)) ⩽ qc 2 Eh 0 (t 0 , ξ)H (t) e−α(t−t0), t ⩾ t 0 , (8)