Exponential Stability of Impulsive Stochastic Delay Differential Systems

Definition 2.1. The trivial solution of system (2.1) is said to be

• (1)

  pth moment exponentially stable if for any initial data $$, the solution x(t) satisfies

  $$ (2.4)
  where λ and C are positive constants independent of t₀,

• (2)

  almost exponentially stable if the solution x(t) satisfies

  $$ (2.5)
  for any initial data $$ and λ > 0.

Definition 2.2. The average impulsive interval of the impulsive sequence {t_k} _k∈ℕ is equal to a positive number T_a if there exists a positive integer N₀ such that

where N(t, t₀) denotes the number of impulsive times of the impulsive sequence {t_k} _k∈ℕ on the interval (t₀, t).

Theorem 3.1. Assume that there exist positive constants c₁, c₂, p, γ₁ such that

•  

  (H₁) c₁ | x|^p ≤ V(t, x) ≤ c₂ | x|^p,

•  

  (H₂) for t ∈ [t_k−1, t_k), k ∈ ℕ,

  $$ (3.1)
  provided $$ satisfying that EV(t + θ, φ(θ)) ≤ qEV(t, φ(0)), θ ∈ [−τ, 0],

•  

  (H₃) there exists a positive constant μ > 1 such that

  $$ (3.2)

•  

  (H₄) $$, T_a > ln μ/γ₁.

Then the trivial solution of system (2.1) is pth moment exponentially stable.

Proof. According to (H₁), we see that

where M = c₂E∥ξ∥^p. We will show that Suppose (3.4) is not true. Then there exist some t ∈ (t₀, t₁) such that $$. Set $$. It follows that t^* ∈ [t₀, t₁) and $$. Moreover, there is a sequence {s_m} _m≥1 and s_m ↓ t^* such that Consequently, which implies that Noticing that the solution x(t) and functionals V, ℒV are continuous on [t^*, t₁), thus we obtain for sufficiently small h > 0. Using Itô’s formula, we derive which yields that This contradicts (3.5). Therefore (3.4) holds. Now, we assume that We will prove that In view of (3.11) and (H₃), we get (3.11) holds for t = t_k. Suppose (3.12) is not true; then, there exist some t ∈ (t_k, t_k+1) such that $$. Seting $$, we have t^* ∈ [t_k, t_k+1) and $$. Moreover, there is a sequence {s_m} _m≥1 and s_m ↓ t^* such that For −τ ≤ θ ≤ 0, there exists an integer j ∈ [0, k] such that t^* + θ ∈ [t_j, t_j+1). Hence Thus, from (H₂), we have Similarly, this can lead to a contradiction, which implies that (3.12) holds. From Definition 2.2, we see that Consequently, where λ = γ₁ − ln μ/T_a > 0. This completes the proof.

Remark 3.2. Theorem 3.1 gives the conditions under which the impulsive disturbances do not destroy the stability of system (2.1). When the impulsive effects are destabilizing, the impulses should not happen too frequently. Therefore, in order to maintain the stability of continuous dynamical system, the average impulsive interval is used to control the impulsive frequency.

Remark 3.3. In Theorem 3.1, the impulses are regarded as disturbance; therefore, the condition μ > 1 is reasonable. It is worth pointing out that in Theorem 3.1, for arbitrary small ɛ and any T_a > 0, the impulsive interval can be less than ɛ and simultaneously the average impulsive intervals are not less than T_a. That is, high-density impulses are allowed to happen in a certain interval, but we need low-density impulses to follow as a compensation.

In the following theorem, when the continuous dynamics in the system (2.1) is unstable, it is shown that the system (2.1) can be stabilized by impulses.

Theorem 3.4. Let β = inf _k∈ℕ{t_k − t_k−1} and there is a positive integer l such that (l − 1)β < τ ≤ lβ. Assume that there exist positive constants c₁, c₂, p, γ₂ such that

•  

  (H₁) c₁ | x|^p ≤ V(t, x) ≤ c₂ | x|^p,

•  

  (H₂) for t ∈ [t_k−1, t_k), k ∈ ℕ,

  $$ (3.18)
  provided $$ satisfying that EV(t, φ(θ)) ≤ qEV(t, φ(0)), θ ∈ [−τ, 0],

•  

  (H₃) there exists a positive constant μ < 1 such that

  $$ (3.19)

•  

  (H₄) qμ^l ≥ 1,   T_a < −ln μ/γ₂.

Then the trivial solution of system (2.1) is pth moment exponentially stable.

Proof. In view of (H₁), we obtain

where $$. We will show that Suppose (3.21) is not true. Then there exist some t ∈ (t₀, t₁) such that $$. Set $$, which yields that $$ for t ∈ [t₀, t^*) and $$. Moreover, there is a sequence {s_m} _m≥1 and s_m ↓ t^* such that Noticing that we derive Since the solution x(t) and functionals V, ℒV are continuous on [t^*, t₁), we see that for sufficiently small h > 0. Using Itô’s formula, we obtain which implies This contradicts (3.22). Thus (3.21) holds. Now, we assume that We will prove that Using (H₂) and (3.28) implies that (3.29) holds for t = t_k. Suppose (3.29) is not true. Then, there exist some t ∈ [t_k, t_k+1) such that $$. Seting $$, we have t^* ∈ [t_k, t_k+1) and $$. Moreover, there is a sequence {s_m} _m≥1 and s_m ↓ t^* such that For −τ ≤ θ ≤ 0, there exists an integer j such that t^* + θ ∈ [t_j, t_j+1), k − l ≤ j ≤ k; then, It follows that Similarly, this can lead to a contradiction, which implies that (3.29) holds.

According to (3.16), we obtain

where λ = −(ln μ/T_a + γ₂) > 0. This completes the proof.

Remark 3.5. Theorem 3.4 shows that an unstable stochastic delay differential system can be successfully stabilized by impulses. The average impulsive interval is used to estimate the impulsive frequency; namely, the impulsive frequency should exceed a lower bound so that there exist enough impulses to stabilize the unstable continuous dynamical system.

Corollary 3.6. Let θ ≡ 0 and τ = 0 in system (2.1). Assume that there exist positive constants c₁, c₂, p, γ₂ such that

•  

  (H₁) c₁|x|^p ≤ V(t, x) ≤ c₂|x|^p,

•  

  (H₂) for t ∈ [t_k−1, t_k), k ∈ ℕ,

  $$ (3.34)

•  

  (H₃) there exists a positive constant μ < 1 such that

  $$ (3.35)

•  

  (H₄) ln μ/T_a + γ₂ < 0.

Then the trivial solution of system (2.1) is pth moment exponentially stable.

Proof. The proof is similar to the proof given in Theorem 3.4, so we omit the detailed proof.

Theorem 3.7. Assume that p ≥ 1, β = inf _k∈ℕ{t_k − t_k−1} and there exists a positive integer l such that (l − 1)β < τ ≤ lβ. Suppose that the conditions in Theorem 3.1 or Theorem 3.4 hold. Moreover, there exists a constant L > 0, such that

Then the trivial solution of system (2.1) is almost surely exponentially stable.

Proof. By Theorem 3.1 or Theorem 3.4, we derive that the trivial solution of system (2.1) is pth moment exponentially stable. Therefore, there exists a positive constant M₁ such that

It is obvious that Combining the Hölder inequality with (3.36) and (3.37) implies that By virtue of the Burkholder-Davis-Gundy inequality, (3.36), and (3.37), we have where C(p) is a positive constant depending on p only. Thanks to (3.36) and (3.37), we see that Substituting (3.39)–(3.41) into (3.38) gives that where M₂ is a positive constant. Then for an arbitrary ɛ ∈ (0, λ) and n ∈ ℕ, we derive Using the Borel-Cantelly lemma, we see that there exists an n₀(ω) such that for almost all ω ∈ Ω, n ≥ n₀(ω), where nτ ≤ t ≤ (n + 1)τ. It follows that Consequently, Let ɛ → 0. Then the result follows.

Example 4.1. Consider an impulsive stochastic delay differential system as follows:

Choosing p = 2, V(t, x) = x², c₁ = c₂ = 1, and q = 4/3 in Theorem 3.1, then we have Seting γ₁ = 0.49, then EℒV(t, x)<−γ₁EV(t, x). It is clear that μ = 1.44, $$, T_a ≥ ln μ/γ₁ = 0.744. For all ɛ > 0, we let t_2k−1 − t_2(k−1) = 1.488, t_2k − t_2k−1 = ɛ, k ∈ ℕ. Thus, by Theorem 3.1 the trivial solution of system (4.1) is pth moment exponential stability. Set ɛ = 0.05, which yields l = 10 in Theorem 3.7. Obviously, for system (4.1), condition (3.36) holds. Then by Theorem 3.7, the trivial solution of system (4.1) is also almost surely exponential stability.

Figure 1 describes the destabilizing impulsive sequence in the system (4.1) when ɛ = 0.05. It can be seen from Figure 2 that the destabilizing impulses do not destroy the stability of system (4.1).

Example 4.2. Consider an impulsive stochastic delay differential system as follows:

Clearly, for system (4.3), condition (3.36) holds. Let p = 2, V(t, x) = x², c₁ = c₂ = 1, and q = 4 in Theorem 3.4. Then we have Seting γ₂ = 4.1, then EℒV(t, x) < γ₂EV(t, x). It follows that μ = 0.25, T_a < −ln μ/γ₂ = 0.338. Thus, we can choose t_2k−1 − t_2(k−1) = 0.2, t_2k − t_2k−1 = 0.45, k ∈ ℕ, which follows T_a = 0.325, l = 1, and qμ^l = 1 ≥ 1. Then by Theorems 3.4 and 3.7, the trivial solution of system (4.3) is pth moment and almost sure exponential stability.