Volume 2013, Issue 1 854743

Research Article

Open Access

Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes

Junhao Hu,

Junhao Hu

College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China scuec.edu.cn

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Xuerong Mao,

Xuerong Mao

Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK strath.ac.uk

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Chenggui Yuan,

Corresponding Author

Chenggui Yuan

[email protected]

orcid.org/0000-0003-0486-5450

Department of Mathematics, Swansea University, Swansea SA2 8PP, UK swansea.ac.uk

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Junhao Hu,

Junhao Hu

College of Mathematics and Statistics, South-Central University for Nationalities, Wuhan 430074, China scuec.edu.cn

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Xuerong Mao,

Xuerong Mao

Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK strath.ac.uk

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Chenggui Yuan,

Corresponding Author

Chenggui Yuan

[email protected]

orcid.org/0000-0003-0486-5450

Department of Mathematics, Swansea University, Swansea SA2 8PP, UK swansea.ac.uk

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First published: 24 May 2013

https://doi.org/10.1155/2013/854743

Citations: 2

Academic Editor: Chuangxia Huang

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Abstract

This paper concerns Razumikhin-type theorems on exponential stability of stochastic differential delay equations with Markovian switching, where the modulating Markov chain involves small parameters. The smaller the parameter is, the rapider switching the system will experience. In order to reduce the complexity, we will “replace” the original systems by limit systems with a simple structure. Under Razumikhin-type conditions, we establish theorems that if the limit systems are pth-moment exponentially stable; then, the original systems are pth-moment exponentially stable in an appropriate sense.

1. Introduction

The stability of time delay systems is a field of intense research [1, 2]. In [2], the global uniform exponential stability independent of time delay linear and time invariant systems subjected to point and distributed delays was studied. Moreover, noise and time delay are often the sources of instability, and they may destabilize the systems if they exceed their limits [3].

Hybrid delay systems driven by continuous-time Markov chains have been used to model many practical systems in which abrupt changes may be experienced in the structure and parameters caused by phenomena such as component failures or repairs. An area of particular interest has been the automatic control of the underlying systems, with consequent emphasis on the analysis of stability of the stochastic models. For systems with time delay, there are two approaches to proving stability that correspond to the conventional Lyapunov stability theory. The first is based on Lyapunov-Krasovski functionals, the second on Lyapunov-Razumikhin functions. The latter one originated with Razumikhin [4] for the ordinary differential delay equation which is called Razumikhin-type theorem and was developed by several people [5]. In his paper, Mao [6] was the first who established a Razumikhin-type theorem for stochastic functional differential equations (SFDEs). Roughly speaking, a Razumikhin-type theorem states that if the derivative of a Lyapunov function along trajectories is negative whenever the current value of the function dominates other values over the interval of time delay; then, the Lyapunov function along trajectories will converge to zero. The Razumikhin methods have been widely used in the study of stability for functional and differential-delay systems. In this work, we shall investigate stochastic differential delay equations with Markovian switching (SDDEwMSs). The switching we shall use will be a finite-state Markov chain, which incorporates various considerations into the models and often results in the underlying Markov chain having a large state space. To overcome the difficulties and to reduce the computational complexity, much effort has been devoted to the modeling and analysis of such systems, in which one of the main ideas is to split a large-scale system into several classes and lumping the states in each class into one state; see [7–9]. Starting from the work [10], by introducing a small parameter ε > 0, a number of asymptotic properties of the Markov chain r^ε(·) have been established. One of the main results in [9] is that a complicated system can be replaced by the corresponding limit system having a much simpler structure. In [11, 12], long-term behavior of SDEwMSs and SDDEwMSs was investigated, respectively, while in [13, 14] the stability of random delay system with two-time-scale Markovian switching was studied. Using the stability of the limit system as a bridge, the desired asymptotic properties of the original system is obtained using perturbed Lyapunov function methods. In this work, we shall establish a Razumikhin-type theorem for SDDEwMSs.

The remainder of this work is organised as follows: in the next section, we shall begin with the formulation of the problem. Section 3 investigates the Razumikhin-type theorem for SDDEs driven by Brownian motion. The exponential stability for SDDEs driven by pure jumps is discussed in Section 4.

2. Preliminaries

Let (Ω, ℱ, {ℱ_t} _t≥0, ℙ) be a complete probability space with a filtration {ℱ_t} _t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous, and ℱ₀ contains all ℙ-null sets). Throughout the paper, we let B(t) = (B₁(t), …, B_m(t)) ^T be an m-dimensional Brownian motion defined on the probability space (Ω, ℱ, {ℱ_t} _t≥0, ℙ). If A is a vector or matrix, its transpose is denoted by A^T. Let |·| denote the Euclidean norm in ℝⁿ as well as the trace norm of a matrix. For τ > 0, C([−τ, 0]; ℝⁿ) denotes the family of continuous functions from [−τ, 0] to ℝⁿ with the norm ∥φ∥ = sup _{−τ≤θ≤0}|φ(θ)|. Denote by the family of all ℱ measurable and bounded C([−τ, 0]; ℝⁿ)-valued random variable. We will denote the indicator function of a set G by I_G.

Let r(t) (t ≥ 0) be a right-continuous Markov chain on (Ω, ℱ, {ℱ_t} _t≥0, ℙ) taking values in a finite state space 𝕊 = {1,2, …, N} with the generator Γ = (γ_ij) _N×N given by

()

where δ > 0 and γ_ij is the transition rate from i to j satisfying γ_ij > 0 if i ≠ j and γ_ii = −∑_i≠j γ_ij. We assume the Markov r(·) is independent of the Brownian motion B(·). It is well known that almost every sample path r(·) is a right-continuous step function with finite number of simple jumps in any finite subinterval of ℝ₊ : = [0, ∞). As a standing hypothesis, we assume that the Markov chain is irreducible. This is equivalent to the condition that for any i, j ∈ 𝕊, we can find i₁, i₂, …, i_k ∈ 𝕊 such that

()

Thus, Γ always has an eigenvalue 0. The algebraic interpretation of irreducibility is rank (Γ) = N − 1. Under this condition, the Markov chain has a unique stationary (probability) distribution πΓ = 0, subject to

and π_j > 0 for all j ∈ 𝕊. For a real valued function σ(·) defined on 𝕊, we define

()

for each κ ∈ 𝕊.

Consider the following stochastic delay system with Markovian swtching:

()

where f : ℝⁿ × ℝⁿ × 𝕊 → ℝⁿ and g : ℝⁿ × ℝⁿ × 𝕊 → ℝ^n×m.

To highlight the fast and slow motions, we introduce a parameter ε > 0 and rewrite the Markov chain r(t) as r^ε(t) and the generator Γ as Γ^ε. Γ^ε is given by

()

where

represents the fast varying motions, and

represents the slowly changing dynamics. We denote

, and

. To the reduction of complexity,

needs to have a certain structure. Suppose that

()

with

and N = N₁ + N₂ + ⋯+N_l, and that

()

where for each k ∈ {1, …, l} and

is a generator of a Markov chain taking values in 𝕊^k. We impose the following hypothesis:

(H1) For each k ∈ {1, …, l}, is irreducible.

To highlight the effect of the fast switching, we rewrite the system (4) as

()

To assure the existence and uniqueness of the solution, we give the following standard assumptions.

(H2) For any integer R, there is a constant h_R > 0, such that

()

for all κ ∈ 𝕊 and those x, x₁, y, y₁ ∈ ℝⁿ with |x | ∨|x₁ | ∨|y | ∨|y₁ | ≤ R.

(H3) There is an h > 0, such that for any x, y ∈ ℝⁿ, κ ∈ 𝕊,

()

Under the assumptions (H2) and (H3), system (8) has a unique solution denoted by x^ε,ξ,ℓ(t) on t ≥ −τ, where the notation x^ε,ξ,ℓ emphasizes the dependence on the initial data (ξ, ℓ). Moreover, for every p > 0 and any compact subset K of C([−τ, 0]; ℝⁿ), there exists a positive constant H which is independent of ε such that

()

We will consider the stability of system (8), but the state space of the Markov chain is large, and it is difficult to handle (8) directly. So we will consider the average system of (8). To proceed, lump the states in each 𝕊^k into a single state and define an aggregated process

()

Denote the state space of

, the stationary distribution

and

. Define

()

with

and

. It has been known that

converges weakly to

as ε → 0, where

is a continuous-time Markov chain with generator

and state space

(cf. [9]).

Define

()

for each s_ij ∈ 𝕊ⁱ with i ∈ {1, …, l} and j ∈ {1, …, N_i}. It is easily seen that

and

are the averages with respect to the stationary distribution of the Markov chain. Note that for any (x, y)≠(0,0), g(x, y, s_ij)g^T(x, y, s_ij) are nonnegative definite matrices, so we find its “square root” of (15), which is denoted by

. For degenerate diffusions, we can see the argument in [15].

The averaged system of (8) is defined as follows:

()

3. Moment Exponential Stability

In this section, we shall establish the Razumikhin-type theorem on the exponential stability for (8).

Let be the class of nonnegative real-valued functions defined on that are p-times continuously differentiable with respect to x. We give the following assumption about for some p ≥ 4.

(H4) For each , V(x, i) → ∞ as |x | → ∞. Moreover, ∂^pV(x, i) = O(1), ∂^ℓV(x, i)(|x|^ℓ+|y|^ℓ) ≤ K(|x|^p+|y|^p + 1) for 1 ≤ ℓ ≤ p − 1, where ∂^ℓV(x, i) denotes the ℓth derivative of V(x, i) with respect to x and O(y) denotes the function of y satisfying sup _y | O(y)|/y < ∞.

Theorem 1. Let (H1)–(H3) hold; there is a function satisfying (H4), and there are positive constants λ, c₁, c₂, and q > 1 such that

(i)
c₁ | x|^p ≤ V(x, i) ≤ c₂ | x|^p,
(ii)
provided ,

where

()

Then, for all ξ ∈ C([−τ, 0]; ℝⁿ),

()

where

()

Remark 2. Note that the conditions of Theorem 1 are sufficient conditions for the average system (16) (or the limit process ). However the conclusion of Theorem 1 is about the process x^ε(t). Since the structure of the the average system (16) is much simpler than that of x^ε(t), this theorem has reduced the computational complexity for the system (8).

Remark 3. limsup_ε→0𝔼 | x^ε(t)|^p does exist by (11).

Proof of Theorem 1. Define

()

Note that

()

We extend r(t) to [−τ, 0] by setting r(t) = r(0); then,

is right continuous on t ≥ −τ.

Let be arbitrary, and define

()

If we can show that U(t) ≤ c₁ν₂, then the proof is completed.

If t ∈ [−τ, 0], by condition (i),

()

If t ≥ 0, we will prove that U(t) ≤ c₁ν₂. Otherwise, there exists the smallest ρ ∈ (0, ∞) such that all t ∈ [−τ, ρ), U(t) ≤ c₁ν₂ and U(ρ) ≥ c₁ν₂ as well as for all suffieciently small .

For t ∈ [ρ − τ, ρ),

()

, then

, t ∈ [ρ − τ, ρ).

Since converges to with probability one (see Lemma 2.3 in [12]), by condition (i), we can derive

()

Recalling the fact

and using the uniqueness of the equation, we then have

, a.e. t > 0. Therefore we have

()

Then U(ρ) = 0, which is a contradiction. Hence we see that

. For t ∈ [ρ − τ, ρ), there exists a q > 1 such that

()

Consequently, there exists a sufficiently small ε₀ > 0, such that, for any ε ∈ (0, ε₀),

()

By condition (ii),

()

then,

()

Noting that

, we have

()

We now consider

()

By the definition of operator ℒ, we have

()

By the definition of

()

This, together with assumption (H2), implies

()

By the argument of Lemma 7.14 in [9], the right side of above inequality is equivalent to to 0; that is, I₂ = 0. Similarly, we can show

()

By the definition of

and

, we have

()

hence

()

By assumption (H4) and the argument of Lemma 7.14 in [9], we have the right side of above inequality is equivalent to 0, that is, I₄ = 0.

Therefore by the condition (ii)

()

this is

()

This contradicts the definition of ρ. The proof is now completed.

Example 4. Let r^ε(·) be a Markov chain generated by Γ^ε given in (5) with

()

The generator

consists of two irreducible blocks. The stationary distributions are μ¹ = (0.5,0.5), μ² = (1/7, 2/7, 4/7), and

()

Consider a one-dimensional equation

()

with

()

Then the limit equation is

()

where

is the Markov chain generated by

and

()

Let V(x, 1) = 2x², V(x, 2) = x²; then,

()

Consequently

()

It is easy to see that we can find a q > 1 such that (1/4)−(q/16) > 0. Therefore, for any

satisfying

on −τ ≤ θ ≤ 0, (49) yields

()

Hence, by Theorem 1, the solution x^ε(t) is mean square stable when ε is sufficient small.

4. Stochastic Delay System with Pure Jumps

In this section we discuss the stability of the following stochastic delay system with pure jumps:

()

where x^ε(t−) = lim _s↑tx^ε(s), b : ℝⁿ × ℝⁿ × 𝕊 × ℝ^m → ℝ^n×m. We assume that the each column b^(l) of the n × m matrix b = [b_ij] depends on z only through the lth coordinate z_l; that is,

()

N(t, z) is a m-dimensional Poisson process, and the compensated Poisson, process is defined by

()

where {N_j, j = 1, …, m} are independent one-dimensional Poisson random measures with characteristic measure {λ_j, j = 1, …, m} coming from m independent one-dimensional Poisson point processes.

The averaged system of (18) is defined as follows:

()

where

. Similar to the definition of

, we define

()

For each s_ij ∈ 𝕊ⁱ with i ∈ {1, …, l} and j ∈ {1, …, N_i}.

To assure the existence and uniqueness of the solution of (52), we also give the following standard assumptions.

(H2′) For any integer R, there is a constant h_R > 0, such that

()

for all i ∈ 𝕊 and those x₁, x₂, y₁, y₂ ∈ ℝⁿ with |x₁ | ∨|x₂ | ∨|y₁ | ∨|y₂ | ≤ R.

(H3′) There is an h > 0, such that for any x, y ∈ ℝⁿ, i ∈ 𝕊,

()

Given V ∈ C^p(ℝⁿ × 𝕊; ℝ₊), we define the operator 𝕃V by

()

where

()

We need the following lemma, for details see [16].

Lemma 5. Let (H1) and (H2′), (H3′) hold, as ε → 0; then, converges weakly to in , where is the space of functions defined on [0, ∞) that are right continuous and have left limits taking values in and endowed with the Skorohod topology.

We now state our main result in this section.

Theorem 6. Let (H1) and (H2′), (H3′) hold; there is a function satisfying (H4), and there are positive constants λ, c₁, c₂, and q > 1 such that

(i)
c₁ | x|^p ≤ V(x, i) ≤ c₂ | x|^p,
(ii)
provided ,

Then, for all ξ ∈ C([−τ, 0]; ℝⁿ),

()

where

()

Proof. As the proof of Theorem 1, define

()

We extend r(t) to [−τ, 0] by setting r(t) = r(0). Then,

is right continuous on t ≥ −τ.

Let be arbitrary, and define

()

If we can show that U(t) ≤ c₁ν₄, then the proof is completed.

If t ∈ [−τ, 0], by condition (i), is the same as the proof of Theorem 1, we have U(t) ≤ c₁ν₄.

In the following we shall prove that U(t) ≤ c₁ν₄ if t ≥ 0. Otherwise, there exists the smallest ρ ∈ (0, ∞) such that all t ∈ [−τ, ρ), U(t) ≤ c₁ν₄, and U(ρ) ≥ c₁ν₄ as well as for all suffieciently small .

As the same in the proof of Theorem 1 we can have that . Hence for t ∈ [ρ − τ, ρ), there exists a q such that

()

Consequently, there exists a sufficiently small ε₀ > 0, such that for any ε ∈ (0, ε₀),

()

By condition (ii),

()

we then have for

()

We now consider

()

By the definition of the operator 𝕃 similar to that of the proof of Theorem 1, we have

()

This implies

()

By the definition of

()

By assumption (H2′), we have

()

By the argument of Lemma 7.14 in [9], the right side of the inequality above is equivalent to 0, that is, J₄ = 0. Similarly, by mean-value theorem, we can show that there exists η^(k)(t) which is between x^ε(t) + b^(k)(x^ε(t−), x^ε((t − τ)−), r^ε(t), z_k) and

such that

()

By the argument of Lemma 7.14 in [9], we have J₃ = 0. Similar to the proof of Theorem 1, we can derive J₂ = 0, J₅ = 0.

Therefore we arrive at

()

then,

()

This contradicts the definition of ρ. The proof is therefore completed.

We shall give an example to illustrate our theory:

Example 7. Let r^ε(·) be a Markov chain generated by

()

here we set

. The stationary distribution is μ = (4/19, 8/19, 3/19, 4/19). Consider a one-dimensional equation

()

with

()

Let

()

Then the limit equation is

()

Let V(x) = x²; then,

()

We can find a q > 1 such that 3 − 2q > 0. Therefore, for any

satisfying

on −τ ≤ θ ≤ 0, (49) yields

()

Hence, by Theorem 6, the solution x^ε(t) is mean square stable.

Acknowledgment

This paper was supported by the National Science Foundation of China with Grant no. 60904005.

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Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes

Abstract

1. Introduction

2. Preliminaries

3. Moment Exponential Stability

4. Stochastic Delay System with Pure Jumps

Acknowledgment

References

Citing Literature

References

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Razumikhin-Type Theorems on Exponential Stability of SDDEs Containing Singularly Perturbed Random Processes

Abstract

1. Introduction

2. Preliminaries

3. Moment Exponential Stability

4. Stochastic Delay System with Pure Jumps

Acknowledgment

References

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