Based on Lyapunov-Krasovskii functional method and stochastic analysis theory, we obtain some new delay-dependent criteria ensuring mean square stability of a class of impulsive stochastic equations. Numerical examples are given to illustrate the effectiveness of the theoretical results.

1. Introduction

It is recognized that the theory of impulsive systems provides a natural framework for the mathematical modeling of many real world phenomena, and impulsive dynamical systems have attracted considerable interest in science and engineering during the past decades. Two classical monographs are Lakshmikantham et al. [1] and Bainov and Simeonov [2]. In general, an impulsive dynamical system can be viewed as a hybrid one comprised of three components: a continuous-time differential equation, which governs the motion of the dynamical systems between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs and a criterion for determining when the states of the systems are to be reset, see Chen and Zheng [3]. Stability properties of impulsive systems have been extensively studied in the literatures. We refer to Li et al. [4, 5], Li et al. [6], Yang [7], Autonio, and Alfonso [8] and the references therein.

Besides impulsive effects, a practical system is usually affected by external stochastic perturbations. Stochastic perturbation is also a factor that makes systems unstable. Recently, stochastic modeling has come to play an important role in many branches of science and industry. An area of particular interest has been stability analysis of impulsive systems with stochastic perturbation. In Yang et al. [9] and Chen et al. [10], the stability properties of nonlinear impulsive stochastic systems are studied using Lyapunov function methods. In Mao et al. [11], a linear matrix inequality approach is proposed for stability analysis of linear uncertain impulsive stochastic systems. However, to the best of our knowledge, there are only few results about this problem.

This paper is inspired by Yang et al. [9], in which the authors considered the problems of stability or robust stabilization for impulsive time delay systems. Unfortunately, they need all the impulsive time sequences to satisfy some strict conditions, that is, the length of the intervals between two jumping time instants must have upper bound or lower bound. But in practical systems, it is always impossible or difficult to obtain it. In this article, by using Lyapunov function methods, together with stochastic analysis, we focus on the mean square stability of trival solution of a class of nonlinear impulsive stochastic time-delay differential systems. We obtain some new conditions ensuring mean square stability of trival solution of the impulsive stochastic differential systems with time-delay. This paper improved some related results.

2. Preliminaries

Throughout this paper, unless explicitly given, for symmetric matrices A and B, the notion A ≥ B (A > B, A ≤ B, A < B) means A − B is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. λ_max(·) (λ_min(·)) represents the maximum (minimum) eigenvalue of the corresponding matrix, respectively. ∥·∥ denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let (Ω, ℱ, {ℱ_t} _t≥0, P) be a complete probability space with a filtration {ℱ_t} _t≥0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC([−τ, 0], ℝⁿ) denote the set of piecewise right continuous function ϕ : [−τ, 0] → ℝⁿ with the norm defined by ∥ϕ∥_τ = sup _{−τ≤θ≤0}∥ϕ(θ)∥, where τ is a known positive constant, PC(δ) = {φ∣φ ∈ PC([−τ, 0], ℝⁿ), ∥φ∥_τ ≤ δ}, denote the family of all ℱ₀-measurable PC([−τ, 0], ℝⁿ)-valued stochastic process φ = {φ(s):−τ ≤ s ≤ 0} with sup _{−τ≤s≤0}𝔼{∥φ(s)∥²} < ∞, where 𝔼{·} represents the mathematical expectation operator with respect to the probability measure P, , ℒ denote the well-known ℒ-operator given by the Itô′s formula.

In this paper, we consider a class of Itô impulsive stochastic differential systems with time delay

(2.1)

where the initial value

, the fixed impulsive time moments t_k satisfy 0 ≤ t₀ < t₁ < t₂ < ⋯<t_k < ⋯(t_k → ∞ as k → ∞). x(t) ∈ ℝⁿ is the system state, f ∈ C(ℝ × ℝⁿ × ℝⁿ, ℝⁿ), g ∈ C(ℝ × ℝⁿ × ℝⁿ, ℝ^n×m). ω(t) ∈ ℝ^m is an standard Brownian motion defined on the complete probability space (Ω, ℱ, {ℱ_t} _t≥0, P). Besides, we assume that H_k(0) = 0, (k = 1,2, …), f(t, 0,0) = 0, g(t, 0,0) = 0 and

(2.2)

In the following, we will divide three cases to consider the mean square stability of system (2.1). We denote by 𝒩_inf(β) and 𝒩_sup(β) the class of impulsive time sequences that satisfy inf _k{t_k − t_k−1} ≥ β and sup _k{t_k − t_k−1} ≤ β, respectively.

We need the following lemma and definitions.

Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]). Assume that f, F ∈ C(G), g ⊂ ℝ² and

(2.3)

If ϕ(t) (t ∈ U₁) and Φ(t) (t ∈ U₂) are the solutions of Cauchy problems

(2.4)

respectively, then for t ∈ (τ, ∞)∩U₁∩U₂,

(2.5)

and for t ∈ (−∞, τ)∩U₁∩U₂,

(2.6)

Definition 2.2. For a given class 𝒩 of admissible impulsive time sequence, the solution of (2.1) is called mean squarely stable if for any ɛ > 0, there exists a scalar δ > 0, such that the initial function implies 𝔼{∥x(t)∥²} < ɛ, t ≥ t₀ for all admissible time sequence in 𝒩.

Definition 2.3 (see Yang et al. [9].)The function V : [t₀ − τ, ∞) × ℝⁿ → ℝ⁺ belongs to class 𝒱^(1,2) if

(1)
the function V(t, x) is continuously differentiable in t and twice continuously differentiable in x on each of the sets [t_k−1, t_k) × ℝⁿ, (k = 1,2, …) and for all t ≥ t₀, V(t, 0) ≡ 0,
(2)
V(t, x) is locally Lipschitaian in x,
(3)
for each k = 1,2, …, there exist finite limits
(2.7)

3. Main Results

Theorem 3.1. Assume that there exist scalars λ₂ > λ₁ > 0, λ_τ > 0, β > 0, λ ≤ 0, ρ > 0 matrix P > 0 and Lyapunov-Krasovskii functional V(t, x(t)) ∈ 𝒱^(1,2), such that

(C1)
,
(C2)
𝔼ℒV(t, x(t)) ≤ λ𝔼V(t, x(t)) + λ_τ𝔼V(t, x_t), t ∈ [t_k−1, t_k), k = 1,2, …, whenever 𝔼V(t, x_t)≤(μ + ρ)𝔼V(t, x(t)),
(C3)
and λ + (μ + ρ)λ_τ ≤ −((ln (μ + ρ))/β),

then the trivial solution of system (2.1) is mean squarely stable over 𝒩_inf(τ + β).

Proof. For any given ɛ > 0, choose . We assume that the initial function and denote the solution x(t, t₀, φ) of system (2.1) through (t₀, φ) by x(t). In the following, we will prove that x(t) is mean square stable over 𝒩_inf(τ + β). For V(t, x(t)) ∈ 𝒱^(1,2), by Itô formula, for t ≠ t_k, k = 1,2, …, we have

(3.1)

where ℒV(t, x(t)) = V_t(t, x(t)) + V_x(t, x(t))f + (1/2)tr (g^TV_xxg).

For t ∈ [t_k−1, t_k), k = 1,2, …, integrate (3.1) from t_k−1 to t, we have

(3.2)

Taking the mathematical expectation of both sides of the above equation, we obtain

(3.3)

So for s ∈ [t, t + Δt] with t + Δt ∈ [t_k−1, t_k) and Δt > 0, if 𝔼V(s, x_s)≤(μ + ρ)𝔼V(s, x(s)), then we have by (C2)

(3.4)

In what follows, we first prove that for t ∈ [t₀ − τ, t₁),

(3.5)

Obviously, for t ∈ [t₀ − τ, t₀], by (C1) and

, we obtain

(3.6)

Now it needs only to prove that for t ∈ (t₀, t₁), (3.5) holds. Otherwise, there exists s ∈ (t₀, t₁), such that

(3.7)

Set

(3.8)

then by (3.6), (3.7), and the continuity of 𝔼V(t, x(t)) on [t₀, t₁), we know that s₁ ∈ (t₀, t₁),

(3.9)

and for t ∈ [t₀ − τ, s₁], (3.5) holds. Set

(3.10)

then by (3.6) and the continuity of 𝔼V(t, x(t)), we have s₂ ∈ [t₀, s₁),

(3.11)

and for t ∈ [s₂, s₁],

(3.12)

which implies with (3.4) and (C3) that for t ∈ [s₂, s₁],

(3.13)

This is a contradiction with (3.9) and (3.11).

Now, we assume that, for t ∈ [t_m−1, t_m), m = 1,2, …, k, (3.5) holds. For m = k + 1, we will show that (3.5) holds. To this end, we first prove that for θ ∈ [−τ, 0],

(3.14)

Noticing [t_k − τ, t_k)⊂[t_k−1, t_k), we assume that there exists some s ∈ [t_k − τ, t_k], such that

(3.15)

then there are two cases to be considered.

(i)
For all t ∈ [t_k−1, s], 𝔼V(t⁻, x(t⁻)) > (λ₁/(μ + ρ))ɛ². Hence, for t ∈ [t_k−1, s], (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1),
(3.16)
this is a contradiction with the assumption.
(ii)
There exists some t ∈ [t_k−1, s), such that 𝔼V(t, x(t)) ≤ (λ₁/(μ + ρ))ɛ². Set
(3.17)
then s₁ ∈ [t_k−1, s),
(3.18)
and for t ∈ [s₁, s], (3.12) and (3.13) hold, which is a contradiction with (3.15) and (3.18), that is, (3.14) holds.

By (2.1), (2.2), and (3.14), we have

(3.19)

Now we will prove that (3.5) holds for t ∈ [t_k, t_k+1). Otherwise, there exists some t ∈ (t_k, t_k+1), such that (3.7) holds. Let

(3.20)

Then by (3.14), (3.19), and the continuity of 𝔼V(t, x(t)) on [t_k, t_k+1), we know that s₁ ∈ (t_k, t_k+1) and 𝔼V(s₁, x(s₁)) = λ₁ɛ². If there exists t ∈ [t_k, s₁], such that 𝔼V(t, x(t)) ≤ (λ₁/μ)ɛ², then let

(3.21)

Otherwise, let s₂ = t_k. Then for t ∈ [s₂, s₁], we obtain (3.12) and (3.13), which follows a contradiction.

By mathematical induction, (3.5) holds for any m = 1,2, …, which implies that system (2.1) is mean squarely stable.

If substituting condition

(C′1) λ₁∥x(t)∥² ≤ V(t, x(t)) ≤ λ₂∥x(t)∥²

for (C1) in Theorem (3.1), then we have the following result.

Theorem 3.2. Assume that there exist scalars λ₂ > λ₁ > 0, λ_τ > 0, λ ≤ 0, ρ > 0, matrix P > 0 and Lyapunov-Krasovskii functional V(t, x(t)) ∈ 𝒱^(1,2), such that conditions (C′1), (C2), and (C3) hold, then the trivial solution of system (2.1) is mean square stable over 𝒩_inf(β).

Proof. The proof is similar to Theorem (3.1), so we omit it. The proof is complete.

Remark 3.3. Comparing the results in Theorems (3.1) and (3.2), we find the influence of the time delay on the mean square stability of system (2.1).

Remark 3.4. When μ > 1, the impulses which may be destabilizing, so we require the impulses should not happen so frequently.

When μ = 1, we have the following results.

Theorem 3.5. Assume that there exist scalars λ₂ > λ₁ > 0, λ_τ > 0, λ ≤ 0, matrix P > 0 and Lyapunov-Krasovskii functional V(t, x(t)) ∈ 𝒱^(1,2), such that condition (C1) and

(C′2)
𝔼ℒV(t, x(t)) ≤ λ𝔼V(t, x(t)) + λ_τ𝔼V(t, x_t), t ∈ [t_k−1, t_k), k = 1,2, … whenever 𝔼V(t, x_t) ≤ 𝔼V(t, x_t),
(C′3)
, λ + λ_τ ≤ 0

hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.

Proof. For any given ɛ > 0, choose . We assume that the initial function . In what follows, we first prove that for t ≥ t₀, (3.5) holds.

Obviously, for t ∈ [t₀ − τ, t₀], by (C1) and , we obtain

(3.22)

Now we should prove that (3.5) holds. Otherwise, there exists s ∈ (t₀, t₁), such that (3.7) holds. By (3.22) and the continuity of 𝔼V(t, x(t)) on [t₀, t₁), we know there exist

and small scalar ρ > 0, such that

(3.23)

and for every

(3.24)

Let

, where ρ₁ > 0 is some scalar. Then [s, s + ρ₁]⊂[t₀, t₁) and for t ∈ [s, s + ρ₁],

(3.25)

which implies with (C′2) and (C′3) that for t ∈ [s, s + ρ₁],

(3.26)

This is a contradiction with the fact 𝔼V(s + ρ₁, x(s + ρ₁)) > 𝔼V(s, x(s)), that is, for t ∈ [t₀ − τ, t₁), (3.5) holds.

Now, we assume that, for t ∈ [t_m−1, t_m), m = 1,2, …, k, (3.5) holds. For m = k + 1, we will show that (3.5) holds. To this end, we first prove that

(3.27)

In fact, by (2.1), (2.2), (C1), and (C′3)

(3.28)

Secondly, we assume that there exists s ∈ (t_k, t_k+1), such that (3.7) holds. By (3.27) and the continuity of 𝔼V(t, x(t)) on [t_k, t_k+1), we know that there exist , ρ₂ > 0 such that for every , , (3.23) and (3.24) hold.

Let , where ρ₂ > 0 is some scalar.

Then for t ∈ [s, s + ρ₂], (3.25) and (3.26) hold. This is a contraction, that is, (3.5) holds for t ∈ [t_k, t_k+1). By mathematical induction, (3.5) holds for any m = 1,2, …, which implies that system (2.1) is mean squarely stable.

Remark 3.6. When μ = 1, both the continuous dynamics and discrete dynamics are stable under the conditions in Theorem (3.5), so the impulse system can be mean squarely stable regardless of how often or how seldom impulses occur.

When μ < 1, we have the following results.

Theorem 3.7. Assume that there exist scalars λ₂ > λ₁ > 0, λ_τ > 0, λ ≤ 0, matrix P > 0, and Lyapunov-Krasovskii functional V(t, x(t)) ∈ 𝒱^(1,2), such that (C1), (C2), and

(C”3)

hold, then

(i)
if 0 < μλ + λ_τ ≤ −λ_τln μ, system (2.1) is mean squarely stable over impulsive time sequences 𝒩_sup(−μln μ/(μλ + λ_τ));
(ii)
if μλ + λ_τ ≤ 0, system (2.1) is mean squarely stable over any impulsive time sequences.

Proof. We prove (i) and omit the proof of (ii).

Because μ < 1 and 0 < μλ + λ_τ ≤ −λ_τln μ, then there exist a sufficiently small ρ₀ > 0, such that

(3.29)

For any given ɛ > 0, choose

. We assume the initial function

. For t ∈ [t₀ − τ, t₀], by (C1), (3.29), and

, we obtain

(3.30)

Now we will prove that (3.5) holds. Otherwise, there exists s ∈ (t₀, t₁), such that (3.7) holds. Set

(3.31)

then by (3.7), (3.30), and the continuity of 𝔼V(t, x(t)) on [t₀, t₁), we know that t^* ∈ (t₀, t₁), 𝔼V(t^*, x(t^*)) = λ₁ɛ². Set

(3.32)

then by (3.30) and the continuity of 𝔼V(t, x(t)), we have

and for

(3.33)

Conditions (C2) and (C”3) imply that for

(3.34)

By Lemma (2.1), (3.29), (3.26), and t₁ − t₀ ≤ (−ln μ/(λ + (λ_τ/μ))), we have

(3.35)

this is a contradiction with the fact 𝔼V(t^*, x(t^*)) = λ₁ɛ².

Now, we assume that, for t ∈ [t_m−1, t_m), m = 1,2, …, k, (3.5) holds. For m = k + 1, we will show that (3.5) holds. To this end, we first prove that

(3.36)

In fact, by (2.1), (2.2), (C1), and (C”3)

(3.37)

Secondly, we assume that there exists s ∈ (t_k, t_k+1), such that (3.7) holds. Set

(3.38)

then by (3.37) and the continuity of 𝔼V(t, x(t)) on [t_k, t_k+1), we have t^* ∈ (t_k, t_k+1),

and 𝔼V(t^*, x(t^*)) = λ₁ɛ²,

On the other hand, for , (3.33) and (3.34) hold, which lead to a contradiction, that is, (3.5) holds for t ∈ [t_k, t_k+1). By mathematical induction, (3.5) holds for any m = 1,2, …, which implies that system (2.1) is mean squarely stable.

4. Application and Numerical Example

As an application, we consider the stochastic impulsive Hopfield neural network with delays in Yang et al. [9] as follows:

(4.1)

where the initial value

, x(t) = (x₁(t), x₂(t), …, x_n(t)) ^T ∈ ℝⁿ is the state vector, C = diag (c₁, c₂, …, c_n), c_i > 0 is the neuron-charging time constant, A = (a_ij) _n×n are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. f(x(t)) = (f₁(x₁(t)), f₂(x₂(t)), …, f_n(x_n(t))) ^T ∈ ℝⁿ and g(x_t) = (g₁(x_1t), g₂(x_2t), …, g_n(x_nt))ℝⁿ, where f_i(x_i(t)) and g_i(x_it) denote, respectively, the measures of response or activation to its incoming potentials of the unit i at time t and time t − τ_i. We also assume that H_k(0) = 0, (k = 1,2, …), f(0) = 0, g(0) = 0, and σ(t, 0,0) = 0, then system (4.1) admits an equilibrium solution x(t) ≡ 0. Moreover, we assume that H(·) satisfies (2.2), and f(·), g(·), σ(·) satisfy

(4.2)

(4.3)

where F, G, K, and K_τ are known constant matrices with appropriate dimensions.

Corollary 4.1. Assume that there exist positive scalars ɛ₁, ɛ₂, β, symmetric matrix P > 0 and . Then the following results hold:

(i)
if μ > 1, λ + μλ_τ < −ln μ/β, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩_inf(τ + β);
(ii)
if μ = 1, λ + λ_τ ≤ 0, then system (4.1) is mean squarely stable over any impulsive time sequence;
(iii)
if μ < 1 and 0 < μλ + λ_τ ≤ −λ_τln μ, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩_sup(−μln μ/(μλ + λ_τ));
(iv)
if μ < 1, μλ + λ_τ ≤ 0, then system (4.1) is mean squarely stable over any impulsive time sequence, where
(4.4)

Remark 4.2. Obviously, for this application, we extended and improved the according results in Yang et al. [9].

By Corollary (4.1), we consider the numerical example in Yang et al. [9].

(4.5)

where t₀ = 0.

Similar to the result, we can verify that the point (0,0) ^T is an equilibrium point and can obtain by calculation that

(4.6)

and

, and, hence, we have μλ + λ_τ = −2.6223, which implies by (iv) in Corollary (4.1) that the above system is mean squarely stable over any impulsive time sequence.

5. Conclusion

In this paper, mean square stability of a class of impulsive stochastic differential equations with time delay has been considered. By Lyapunov-Krasovakii function and stochastic analysis, we obtain some new criteria ensuring mean square stability of the system (2.1). Some related results in Chen and Zheng [3] and Yang et al. [9] have been improved.

Acknowledgment

This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.

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Mean Square Stability of Impulsive Stochastic Differential Systems

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Application and Numerical Example

5. Conclusion

Acknowledgment

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Mean Square Stability of Impulsive Stochastic Differential Systems

Abstract

1. Introduction

2. Preliminaries

3. Main Results

4. Application and Numerical Example

5. Conclusion

Acknowledgment

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