Abstract
In this paper, the exponential stability of a stochastic delay system with impulsive signal is considered, and stability theorem of this system is proposed based on the Lyapunov–Razumikhin method; the convergence rate is also given, which gives theoretical foundation to chaos control and synchronization using the impulsive method. Finally, the classic delay chaos system with white noise and impulsive signal is employed to verify the feasibility and effectiveness of our theorem.
Highlights
The Lyapunov–Razumikhin method is employed to study the exponential stability of impulsive stochastic delay differential equations, and judgement theorem of this problem will be deduced too
In order to show the validity and effectiveness of eorem 1, we investigate chaos controllability of the Lorenz system with white noise and delay using the impulsive method in the sense of exponential stability
In the impulsive control scheme, signals are added to system (28) at discrete time τi, i 1, 2, . . . , and system (28) suffers sudden changes at these instants, and this impulsive control scheme can be written as
Summary
Mathematical Problems in Engineering are first-order continuously differentiable with respect to the first variable and second-order differentiable with respect to the second variable. ‖ψ‖τ sup− τ≤s≤0E‖ψ(s)‖, for ψ ∈ PC([− τ, 0], Rn), τ > 0, where E(X) means the expectation of X and PC([− τ, 0], Rn) denotes the family of all piecewise right continuous functions. 3. Exponential Stability for Impulsive Stochastic Delay Differential Equations Based on Lyapunov–Razumikhin Method. The Lyapunov–Razumikhin method is employed to study the exponential stability of impulsive stochastic delay differential equations, and judgement theorem of this problem will be deduced too. We will show that (6) holds for k 1, i.e., E(V(t)) ≤ ME X0 pτ e− λ(t1− t0), t ∈ t0, t1. Suppose that (6) holds for k m, m ∈ N, i.e., E(V(t)) ≤ ME X0 pτ e− λ(tk− t0), t ∈ tk− 1, tk, (16). From condition (III) and (16), we know that E V tm ≤ dmE V t−m < e− λαe− (λ tm+1− tm)ME X0 pτ e− λ(tm− t0). We will prove that (16) holds for k m + 1, i.e., E(V(t)) ≤ ME X0 pτ e− (λ tm+1− t0), t ∈ tm, tm+1. N, where M∗ max{1, (M/ trivial solution of system (1) is exponentially stable, and the convergence rate is (λ/p). □
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