Stability of uncertain delay differential equations

Abstract

Uncertain delay differential equation is a type of differential equations driven by a canonical Liu process. This paper mainly focuses on the stability of uncertain delay differential equations. At first, the concept of stability in measure, stability in mean and stability in moment for uncertain delay differential equations will be presented. In addition, the sufficient condition for uncertain delay differential equations being stable in measure, in mean and in moment will be derived. Finally, this paper will discuss the relationship among stability in measure, stability in mean and stability in moment.

Keywords

Uncertain delay differential equation stability uncertainty theory

1 Introduction

Probability theory has been used to model indeterminacy phenomena for a long time. To further describe the irregular movement of the pollen in the liquid, Wiener [15] designed a stochastic process called Wiener process. Based on Wiener process, Ito [6] founded stochastic calculus to deal with the integral and differential of a stochastic process in 1940s. Following that, stochastic differential equation was proposed and applied to many fields.

As we know, a premise of applying probability theory is that the available probability distribution is close enough to the true frequency. However, due to various reasons, we sometimes have little or no sample data and it is almost impossible to obtain an exact estimation computed by probability theory. As the result, belief degrees provided by some domain experts might be employed to estimate the distribution. In order to deal with belief degree mathematically, Liu [7] established uncertainty theory in 2007, which is different from probability theory. To represent the quantity with uncertainty, Liu [7] gave the concept of uncertain variable. Furthermore, for describing the evolution of an uncertain phenomenon, Liu [8] proposed the concept of uncertain process, and designed a canonical Liu process [9] in contrast to Wiener process. Meanwhile, Liu [9] founded uncertain calculus to deal with the integral and differential of an uncertain process with respect to the canonical Liu process.

As an important tool to deal with uncertain dynamic systems, uncertain differential equation which is driven by a canonical Liu process, was first proposed by Liu [8] in 2008. In recent years, many researchers have done a lot of work in theory and application about uncertain differential equations. In 2010, Chen and Liu [2] proved a sufficient condition for the existence and uniqueness of the solution of an uncertain differential equation. Simultaneously, they gave an analytic solution for linear uncertain differential equations in [2]. Noting that analytic methods often fail to solve an uncertain differential equation, some numerical methods for solving uncertain differential equation were proposed, such as Euler method [18], Adams-Simpson method [13]. In addition, just like the ordinary differential equation and the stochastic differential equation, stability analysis of the solution is also a central problem in uncertain differential equations. The first concept of stability of uncertain differential equation was presented by Liu [9], and some stability theorems were proved by Yao et al. [19]. After that, different types of stability of uncertain differential equations were explored, such as stability in moment [12] and almost sure stability [10]. In recent years, uncertain differential equations were applied successfully to the financial field, and uncertain stock model [11, 17], uncertain interest rate model [3, 20] and uncertain currency model [14] have emerged. Besides, uncertain differential equations have also been applied to uncertain optimal control [21], and uncertain differential game [16] and so on.

In many practical system models, we used to assume that the future state of the system is determined solely by the present. However, some real phenomena depend not only on the current state of the system but also upon the previous states. Sticking with uncertain differential equation to model is inappropriate in this case. Uncertain delay differential equation is just a good tool to establish a mathematical formulation for such systems. Until now, researchers only explored the existence and uniqueness of solutions for uncertain delay differential equations. In 2010, Barbacioru [1] first proved a local existence and uniqueness result for a special type of uncertain delay differential equations (UDDE). Ge and Zhu [4] presented another existence and uniqueness theorem of solution for UDDE in the finite domain in 2012. In this paper, we will discuss the stability of the uncertain delay differential equation, and provide some theorems for it.

The remainder of this paper is organized as follows. Section 2 is intended to introduce some basic concepts and theorems about uncertainty theory and uncertain delay differential equation used in the later sections. In Sections 3–5, the concepts of stability in measure, stability in mean and stability in moment for uncertain delay differential equation are presented, and some stability theorems are proved. After that, we analyze the relationship among stability in measure, stability in mean and stability in moment in Section 6. Finally, we make a brief conclusion in Section 7.

2 Preliminary

In this section, we will introduce some basic concepts and theorems about uncertainty theory, then introduce uncertain delay differential equation.

2.1 Uncertainty theory

In this section, we review some basic concepts in uncertainty theory, including uncertainty space, uncertain variable, expected value and moment.

Definition 2.1. (Liu [7, 9]) Let = eusm10Mbox = eusm10L be a σ-algebra on a nonempty set Γ. A set function = eusm10M is called an uncertain measure if it satisfies the following axioms:

Axiom 1. (Normality Axiom) = eusm10M {Γ} =1;

Axiom 2. (Duality Axiom) = eusm10M {Λ} + = eusm10M {Λ^c} =1;

Axiom 3. (Subadditivity Axiom) For every countable sequence of {Λ_i} ∈ = eusm10Mbox = eusm10L, we have $= eusm 10 M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} = eusm 10 M {Λ_{i}} .$

Axiom 4. (Product Axiom) Let (Γ_k, = eusm10Mbox = eusm10L_k, = eusm10M_k) be uncertainty spaces for k = 1, 2, ⋯ The product uncertain measure = eusm10M is an uncertain measure satisfying $= eusm 10 M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} {= eusm 10 M}_{k} {Λ_{k}}$ where Λ_k are arbitrarily chosen events from = eusm10Mbox = eusm10L_k for k = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [7]) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, = eusm10Mbox = eusm10L, = eusm10M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ ∈ B} = {γ ∈ Γ|ξ (γ) ∈ B} is an event.

Definition 2.3. (Liu [9]) Let ξ be an uncertain variable. The expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} = eusm 10 M {ξ \geq r} d r - \int_{- \infty}^{0} = eusm 10 M {ξ \leq r} d r$ provided that at least one of the above two integrals is finite.

Definition 2.4. (Liu [7]) Let ξ be an uncertain variable and let p be a positive integer. Then E [ξ^p] is called the p-th moment of ξ.

Theorem 2.1. (Liu [7]) Let ξ be an uncertain variable. Then for any positive numbers x and p, we have $= eusm 10 M {| ξ | \geq x} \leq \frac{E [| ξ |^{p}]}{x^{p}} .$

2.2 Uncertain delay differential equation

In order to model the evolution of uncertain phenomena, an uncertain process was proposed by Liu [8] as a sequence of uncertain variables driven by time. Following that, Liu [9] designed a canonical Liu process which is one of the most important uncertain processes.

Definition 2.5. (Liu [9]) An uncertain process C_t (t ≥ 0) is called a canonical Liu process if

C₀ = 0 and almost all sample paths are Lipschitz continuous,

C_t is a stationary independent increment process,

every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t².

Based on the canonical Liu process, uncertain integral is defined as an counterpart of Ito integral as follows:

Definition 2.6. (Liu [9]) Let X_t be an uncertain process and let C_t be a canonical Liu process. For any partition of the closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = sup_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$

Then the uncertain integral of X_t with respect to C_t is defined by $\int_{a}^{b} X_{t} d C_{t} = lim_{Δ \to 0} \sum_{i = 1}^{k} X_{t_{i}} \cdot (C_{t_{i + 1}} - C_{t_{i}})$ provided that the limit exists almost surely and is finite.

Definition 2.7. (Barbacioru [1]) Let C_t be a canonical Liu process, and f and g are two real functions. $d X_{t} = f (t, X_{t}, X_{t - τ}) d t + g (t, X_{t}, X_{t - τ}) d C_{t}$ (1) is called an uncertain delay differential equation, where τ > 0 is called time delay.

The uncertain delay differential Equation (1) is equivalent to the uncertain delay integral equation

$\begin{matrix} X_{t} & = & X_{0} + \int_{0}^{t} f (s, X_{s}, X_{s - τ}) d s \\ + \int_{0}^{t} g (s, X_{s}, X_{s - τ}) d C_{s} \end{matrix}$ (2)

Theorem 2.2. (Chen and Liu [2]) Suppose C_t is a canonical Liu process, and X_t is an integrable uncertain process on [a, b] with respect to t. Then the inequality $| \int_{a}^{b} X_{t} (γ) d C_{t} (γ) | \leq K (γ) \int_{a}^{b} | X_{t} (γ) | d t$ holds, where K (γ) is the Lipschitz constant of C_t (γ).

Theorem 2.3. (Yao et al. [19]) Let C_t be a canonical Liu process on uncertainty space (Γ, = eusm10Mbox = eusm10L, = eusm10M). Then there exists an uncertain variable K such that K (γ) is a Lipschitz constant of the sample path C_t (γ) for each γ, $lim_{x \to + \infty} = eusm 10 M {γ \in Γ | K (γ) \leq x} = 1$ and $= eusm 10 M {γ \in Γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$

3 Stability in measure

In this section, we will give a definition of stability in measure for an uncertain delay differential equation, and give a sufficient condition for an uncertain delay differential equation being stable inmeasure.

Definition 3.1. The uncertain delay differential Equation (1) is said to be stable in measure if for any two solutions X_t and Y_t with different initial states, we have $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} = eusm 10 M {| X_{t} - Y_{t} | > ε} = 0, \forall t > 0$ for any given number ε > 0.

Theorem 3.1. Assume the uncertain delay differential Equation (1) has a unique solution for each given initial state. Then it is stable in measure if the coefficients f (t, x, y) and g (t, x, y) satisfy the strong Lipschitz condition

$\begin{matrix} | f (t, x_{1}, y) - f (t, x_{2}, y) | \\ \lor | g (t, x_{1}, y) - g (t, x_{2}, y) | \\ \leq L_{t} | x_{1} - x_{2} |, \forall x_{1}, x_{2}, y \in R, t > 0 \end{matrix}$ (3) where L_t is a bounded function satisfying $\int_{0}^{+ \infty} L_{t} d t < + \infty .$

Proof. Assume that X_t and Y_t are two solutions of the uncertain delay differential Equation (1) with different initial states φ (t) and Φ (t) (t ∈ [- τ, 0]), respectively. That is, ${\begin{matrix} d X_{t} = f (t, X_{t}, X_{t - τ}) d t + g (t, X_{t}, X_{t - τ}) d C_{t}, t > 0 \\ X_{t} = φ (t), - τ \leq t \leq 0 \end{matrix}$ and ${\begin{matrix} d Y_{t} = f (t, Y_{t}, Y_{t - τ}) d t + g (t, Y_{t}, Y_{t - τ}) d C_{t}, t > 0 \\ Y_{t} = Φ (t), - τ \leq t \leq 0 . \end{matrix}$

Then for a Lipschitz continuous sample C_t (γ), we have ${\begin{matrix} X_{t} (γ) = X_{0} + \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) d C_{s} (γ), t > 0 \\ X_{t} = φ (t), - τ \leq t \leq 0 \end{matrix}$ and ${\begin{matrix} Y_{t} (γ) = Y_{0} + \int_{0}^{t} f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s \\ + \int_{0}^{t} g (s, Y_{s} (γ), Y_{s - τ} (γ)) d C_{s} (γ), t > 0 \\ Y_{t} = Φ (t), - τ \leq t \leq 0 . \end{matrix}$

By the strong Lipschitz conditions and Theorem 2.2, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | X_{0} - Y_{0} \\ + \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) - f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) - g (s, Y_{s} (γ), Y_{s - τ} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + | \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) - f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s | \\ + | \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) - g (s, Y_{s} (γ), Y_{s - τ} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} L_{s} | X_{s} (γ) - Y_{s} (γ) | d s \\ + K (γ) \int_{0}^{t} L_{s} | X_{s} (γ) - Y_{s} (γ) | d s \\ = | X_{0} - Y_{0} | + (1 + K (γ)) \int_{0}^{t} L_{s} | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$

where K (γ) is the Lipschitz constant of C_t (γ).

It follows from the Gronwall’s inequality [5] that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp ((1 + K (γ)) \int_{0}^{t} L_{s} d s) \\ \leq | X_{0} - Y_{0} | exp ((1 + K (γ)) \int_{0}^{+ \infty} L_{s} d s) \\ \leq sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | exp ((1 + K (γ)) \int_{0}^{+ \infty} L_{s} d s) . \end{matrix}$

Thus we have $\begin{matrix} | X_{t} - Y_{t} | \\ \leq sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | exp ((1 + K (γ)) \int_{0}^{+ \infty} L_{s} d s) \end{matrix}$ almost surely, where K is a nonnegative uncertain variable such that $lim_{x \to + \infty} = eusm 10 M {γ \in Γ | K (γ) \leq x} = 1$ by Theorem 2.3.

Then there exists a positive number H such that $= eusm 10 M {γ | K (γ) \leq H} \geq 1 - ɛ$ for any given ɛ > 0. Take $δ = exp (- (1 + H) \int_{0}^{+ \infty} L_{s} d s) ε .$

Then |X_t (γ) - Y_t (γ) | ≤ ε provided that $sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \leq δ$ and K (γ) ≤ H.

It means $= eusm 10 M {| X_{t} - Y_{t} | \leq ε} > 1 - ɛ, \forall t > 0$ as long as $sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0 .$

In other words, $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} = eusm 10 M {| X_{t} - Y_{t} | > ε} = 0, \forall t > 0 .$

So the uncertain delay differential Equation (1) is stable in measure. □

Remark 3.1. From the sufficient condition (3), it is not difficult to find that the delay term X_t-τ does not work for the stability of uncertain delay differential Equation (1). On one hand, we can obtain this conclusion from the proof of Theorem 3.1. On the other hand, we can analyze the reason from the following perspective: the current term X_t and the delay term X_t-τ have an hidden relationship and restrict mutually from (1). That is, once the information of X_t determined, the delay term X_t-τ will be determined. Hence, we can establish constraints using the information from one aspect.

Remark 3.2. Theorem 3.1 provides the sufficient condition but not the necessary condition for uncertain delay differential equation being stable in measure.

Example 3.1. Since most uncertain delay differential equations have no analytical solution, this paper considers a simple uncertain delay differential equation $d X_{t} = a X_{t - τ} d t + σ d C_{t}, t \in [0, + \infty),$ (4) which has the analytical solution.

Obviously, the coefficients of (4) satisfy the strong Lipschitz condition in Theorem 3.1. In what follows, we investigate the stability in measure of the solution of (4).

The analytical solution of (4) with two initial states φ (t) and Φ (t) (t ∈ [- τ, 0]) is $X_{t} = {\begin{matrix} φ (t), t \in [- τ, 0] \\ φ (0) + a \int_{0}^{t} φ (s - τ) d s + σ C_{t}, t \in [0, τ] \\ X_{τ} + a \int_{τ}^{t} X_{s - τ} d s + σ (C_{t} - C_{τ}), t \in [τ, 2 τ] \\ \dots \dots \end{matrix}$ and $Y_{t} = {\begin{matrix} Φ (t), t \in [- τ, 0] \\ Φ (0) + a \int_{0}^{t} Φ (s - τ) d s + σ C_{t}, t \in [0, τ] \\ Y_{τ} + a \int_{τ}^{t} Y_{s - τ} d s + σ (C_{t} - C_{τ}), t \in [τ, 2 τ] \\ \dots \dots \end{matrix}$ respectively.

Then $\begin{matrix} | X_{t} - Y_{t} | \\ = {\begin{matrix} | φ (t) - Φ (t) |, t \in [- τ, 0] \\ | φ (0) - Φ (0) | + a \int_{0}^{t} | φ (s - τ) \\ - Φ (s - τ) | d s, t \in [0, τ] \\ | X_{τ} - Y_{τ} | + a \int_{τ}^{t} | X_{s - τ} - Y_{s - τ} | d s, t \in [τ, 2 τ] \\ \dots \dots \end{matrix} \end{matrix}$

Therefore, $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} = eusm 10 M {| X_{t} - Y_{t} | > ε} = 0, \forall t > 0$ for any ε > 0, and the uncertain delay differential Equation (4) is stable in measure by Definition 3.1.

4 Stability in mean

In this section, we mainly investigate the stability in mean for uncertain delay differential equations.

Definition 4.1. The uncertain delay differential Equation (1) is said to be stable in mean if for any two solutions X_t and Y_t with different initial states, we have $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [| X_{t} - Y_{t} |] = 0, \forall t > 0 .$

Theorem 4.1. Assume the uncertain delay differential Equation (1) has a unique solution for each given initial state. Then it is stable in mean if the coefficients f (t, x, y) and g (t, x, y) satisfy the strong Lipschitz condition $\begin{matrix} | f (t, x_{1}, y) - f (t, x_{2}, y) | & \leq & F_{t} | x_{1} - x_{2} |, \\ | g (t, x_{1}, y) - g (t, x_{2}, y) | & \leq & G_{t} | x_{1} - x_{2} |, \\ (\forall x_{1}, x_{2}, y & \in & R, t > 0) \end{matrix}$ where F_t and G_t are two bounded functions satisfying $\int_{0}^{+ \infty} F_{t} d t < + \infty, \int_{0}^{+ \infty} G_{t} d t < \frac{π}{\sqrt{3}} .$

By the strong Lipschitz conditions and Theorem 2.2, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | X_{0} - Y_{0} \\ + \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) \\ - f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) - g (s, Y_{s} (γ), \\ Y_{s - τ} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + | \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) \\ - f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s | \\ + | \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) - g (s, Y_{s} (γ), \\ Y_{s - τ} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} F_{s} | X_{s} (γ) - Y_{s} (γ) | d s \\ + K (γ) \int_{0}^{t} G_{s} | X_{s} (γ) - Y_{s} (γ) | d s \\ = | X_{0} - Y_{0} | \\ + \int_{0}^{t} (F_{s} + K (γ) G_{s}) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$ where K (γ) is the Lipschitz constant of C_t (γ).

It follows from the Gronwall’s inequality [5] that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{t} F_{s} d s) \\ exp (\int_{0}^{t} K (γ) G_{s} d s) \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} F_{s} d s) \\ exp (\int_{0}^{+ \infty} K (γ) G_{s} d s) \\ \leq sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | exp (\int_{0}^{+ \infty} F_{s} d s) \\ exp (\int_{0}^{+ \infty} K (γ) G_{s} d s) . \end{matrix}$

Thus we have

$\begin{matrix} | X_{t} - Y_{t} | \leq sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \\ exp (\int_{0}^{+ \infty} F_{s} d s) exp (K \int_{0}^{+ \infty} G_{s} d s) \end{matrix}$ (5) almost surely, where K is a nonnegative uncertain variable such that $\begin{matrix} = eusm 10 M {γ \in Γ | K (γ) \geq x} \\ = 1 - = eusm 10 M {γ \in Γ | K (γ) \leq x} \\ \leq 1 - (2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1) \\ = 2 {(1 + exp (\frac{π x}{\sqrt{3}}))}^{- 1} . \end{matrix}$

Taking expected value on both sides of expression (5), we have $\begin{matrix} E [| X_{t} - Y_{t} |] \leq sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \\ exp (\int_{0}^{+ \infty} F_{s} d s) E [exp (K \int_{0}^{+ \infty} G_{s} d s)] . \end{matrix}$

Since $\int_{0}^{+ \infty} F_{s} d s < + \infty,$ we immediately have $exp (\int_{0}^{+ \infty} F_{s} d s) < + \infty .$

Since $\int_{0}^{+ \infty} G_{s} d s < \frac{π}{\sqrt{3}},$ it follows from definition of expected value that $\begin{matrix} E [exp (K \int_{0}^{+ \infty} G_{s} d s)] \\ = \int_{0}^{+ \infty} = eusm 10 M {γ | exp (K (γ) \int_{0}^{+ \infty} G_{s} d s) \geq x} d x \\ = \int_{0}^{+ \infty} = eusm 10 M {γ | K (γ) \geq \frac{ln x}{\int_{0}^{+ \infty} G_{s} d s}} d x \\ \leq 2 \int_{0}^{+ \infty} \frac{1}{1 + x^{π / (\sqrt{3} \int_{0}^{+ \infty} G_{s} d s)}} d x < + \infty . \end{matrix}$

Hence, we have $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [| X_{t} - Y_{t} |] = 0, \forall t > 0$ and the uncertain delay differential Equation (1) is stable in mean. □

Remark 4.1. Theorem 4.1 provides the sufficient condition but not the necessary condition for uncertain delay differential equation being stable in mean.

Example 4.1. Consider the uncertain delay differential Equation (4) in Example 3.1. Obviously, the coefficients of (4) satisfy the strong Lipschitz condition in Theorem 4.1. It follows from the conclusion in Example 3.1, we have $\begin{matrix} lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [| X_{t} - Y_{t} |] \\ = lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} | X_{t} - Y_{t} | \\ = 0 . \end{matrix}$

According to Definition 4.1, uncertain delay differential Equation (4) is stable in mean.

5 Stability in moment

In this section, we first give a definition of stability in p-th moment for uncertain delay differential equations, and then present a sufficient condition for uncertain delay differential equations being stable in p-th moment, where p is a positive number. Besides, we discuss the relationship between stability in p₁-th moment and stability in p₂-th moment, where p₁ and p₂ are positive numbers.

Definition 5.1. The uncertain delay differential Equation (1) is said to be stable in p-th moment (0 < p < + ∞) if for any two solutions X_t and Y_t with different initial states, we have $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [{(| X_{t} - Y_{t} |)}^{p}] = 0, \forall t > 0 .$

Remark 5.1. In particular, when p = 1, it degenerates into stable in mean.

Theorem 5.1. Assume the uncertain delay differential Equation (1) has a unique solution for each given initial state. Then it is stable in p-th moment (0 < p < + ∞) if the coefficients f (t, x, y) and g (t, x, y) satisfy the strong Lipschitz condition $\begin{matrix} | f (t, x_{1}, y) - f (t, x_{2}, y) | & \leq & F_{t} | x_{1} - x_{2} |, \\ | g (t, x_{1}, y) - g (t, x_{2}, y) | & \leq & G_{t} | x_{1} - x_{2} |, \\ (\forall x_{1}, x_{2}, y & \in & R, t > 0) \end{matrix}$ where F_t and G_t are two bounded functions satisfying $\int_{0}^{+ \infty} F_{t} d t < + \infty, \int_{0}^{+ \infty} G_{t} d t < \frac{π}{\sqrt{3} p} .$

Proof. Assume that X_t and Y_t are two solutions of the uncertain delay differential Equation (1) with different initial states φ (t) and Φ (t) (t ∈ [- τ, 0]), respectively. That is, ${\begin{matrix} d X_{t} = f (t, X_{t}, X_{t - τ}) d t \\ + g (t, X_{t}, X_{t - τ}) d C_{t}, t > 0 \\ X_{t} = φ (t), - τ \leq t \leq 0 \end{matrix}$ and ${\begin{matrix} d Y_{t} = f (t, Y_{t}, Y_{t - τ}) d t + g (t, Y_{t}, Y_{t - τ}) d C_{t}, t > 0 \\ Y_{t} = Φ (t), - τ \leq t \leq 0 . \end{matrix}$

By the strong Lipschitz conditions and Theorem 2.2, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | X_{0} - Y_{0} \\ + \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) \\ - f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s \\ + \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) \\ - g (s, Y_{s} (γ), Y_{s - τ} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \\ + | \int_{0}^{t} f (s, X_{s} (γ), X_{s - τ} (γ)) \\ - f (s, Y_{s} (γ), Y_{s - τ} (γ)) d s | \\ + | \int_{0}^{t} g (s, X_{s} (γ), X_{s - τ} (γ)) \\ - g (s, Y_{s} (γ), Y_{s - τ} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} F_{s} | X_{s} (γ) - Y_{s} (γ) | d s \\ + K (γ) \int_{0}^{t} G_{s} | X_{s} (γ) - Y_{s} (γ) | d s \\ = | X_{0} - Y_{0} | \\ + \int_{0}^{t} (F_{s} + K (γ) G_{s}) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$ where K (γ) is the Lipschitz constant of C_t (γ).

Thus we have

$\begin{matrix} | X_{t} - Y_{t} | \leq sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \\ exp (\int_{0}^{+ \infty} F_{s} d s) exp (K \int_{0}^{+ \infty} G_{s} d s) \end{matrix}$ (6) almost surely, where K is a nonnegative uncertain variable such that $= eusm 10 M {γ \in Γ | K (γ) \geq x} \leq 2 {(1 + exp (\frac{π x}{\sqrt{3}}))}^{- 1}$ by Theorem 2.3. Taking p-th moment on both sides of expression (6), we have $\begin{matrix} E [{(| X_{t} - Y_{t} |)}^{p}] \\ \leq {(sup_{s \in [- τ, 0]} | X_{s} - Y_{s} |)}^{p} exp (p \int_{0}^{+ \infty} F_{s} d s) \\ E [exp (pK \int_{0}^{+ \infty} G_{s} d s)] . \end{matrix}$

Since $\int_{0}^{+ \infty} F_{s} d s < + \infty,$ we immediately have $exp (p \int_{0}^{+ \infty} F_{s} d s) < + \infty .$

Since $\int_{0}^{+ \infty} G_{s} d s < \frac{π}{\sqrt{3} p},$ it follows from definition of expected value that $\begin{matrix} E [exp (pK \int_{0}^{+ \infty} G_{s} d s)] \\ = \int_{0}^{+ \infty} = eusm 10 M {γ | exp (K (γ) p \int_{0}^{+ \infty} G_{s} d s) \geq x} d x \\ = \int_{0}^{+ \infty} = eusm 10 M {γ | K (γ) \geq \frac{ln x}{p \int_{0}^{+ \infty} G_{s} d s}} d x \\ \leq 2 \int_{0}^{+ \infty} \frac{1}{1 + x^{π / (p \sqrt{3} \int_{0}^{+ \infty} G_{s} d s)}} d x < + \infty . \end{matrix}$

Hence, we have $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [{(| X_{t} - Y_{t} |)}^{p}] = 0, \forall t > 0$ and the uncertain delay differential Equation (1) is stable in p-th moment. □

Remark 5.2. Theorem 5.1 gives the sufficient condition but not the necessary condition for uncertain delay differential equation being stable in p-th moment.

Example 5.1. Consider the uncertain delay differential Equation (4) in Example 3.1. It is obvious that the coefficients of Equation (4) satisfy the strong Lipschitz condition in Theorem 5.1. It follows from the conclusion in Example 3.1, we have $\begin{matrix} lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [{(| X_{t} - Y_{t} |)}^{p}] \\ = lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} {(| X_{t} - Y_{t} |)}^{p} = 0, \forall t > 0 \end{matrix}$ for any finite positive p.

By Definition 5.1, uncertain delay differential Equation (4) is stable in p-th moment.

Theorem 5.2. For any two real numbers p₁ and p₂ (0 < p₁ < p₂ < + ∞), if the uncertain delay differential Equation (1) is stable in p₂-th moment, then it is stable in p₁-th moment.

Proof. Assume that X_t and Y_t are two solutions of the uncertain delay differential Equation (1) with different initial states φ (t) and Φ (t) (t ∈ [- τ, 0]), respectively. Then it follows from the definition of stability in p₂-th moment that $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [{(| X_{t} - Y_{t} |)}^{p_{2}}] = 0, \forall t > 0 .$

By Hölder’s inequality, we have $\begin{matrix} E [{(| X_{t} - Y_{t} |)}^{p_{1}}] \\ = E [{(| X_{t} - Y_{t} |)}^{p_{1}} \cdot 1] \\ \leq \sqrt[p_{2} / p_{1}]{E [{(| X_{t} - Y_{t} |)}^{p_{1} \cdot p_{2} / p_{1}}]} \\ \cdot \sqrt[p_{2} / (p_{2} - p_{1})]{E [1^{p_{2} / (p_{2} - p_{1})}]} \\ = \sqrt[p_{2} / p_{1}]{E [{(| X_{t} - Y_{t} |)}^{p_{2}}]}, \forall t > 0 . \end{matrix}$

Thus stability in p₂-th moment implies stability in p₁-th moment when p₁ < p₂. □

6 Comparisons

In the previous sections, we have found that stability in mean is equivalent to stability in 1-th moment. In this section, we mainly analyze the relationship among stability in measure, stability in mean as well as stability in moment for an uncertain delay differential equation.

Theorem 6.1. If the uncertain delay differential Equation (1) is stable in mean, then it is stable in measure.

Proof. Assume that X_t and Y_t are two solutions of the uncertain delay differential Equation (1) with different initial states φ (t) and Φ (t) (t ∈ [- τ, 0]), respectively. Then it follows from the definition of stability in mean that $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [| X_{t} - Y_{t} |] = 0, \forall t > 0 .$

Then for any given real number ε ≥ 0, we have $\begin{matrix} lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} = eusm 10 M {| X_{t} - Y_{t} | \geq ε} \\ \leq lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [| X_{t} - Y_{t} |] / ε = 0, \forall t > 0 \end{matrix}$ by Theorem 2.1. Thus stability in mean implies the stability in measure. □

Theorem 6.2. If the uncertain delay differential Equation (1) is stable in p-th moment (0 < p < + ∞), then it is stable in measure.

Proof. Assume that X_t and Y_t are two solutions of the uncertain delay differential Equation (1) with different initial states φ (t) and Φ (t) (t ∈ [- τ, 0]), respectively. Then it follows from the definition of stability in p-th moment that $lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [{(| X_{t} - Y_{t} |)}^{p}] = 0, \forall t > 0 .$

Then for any given real number ε ≥ 0, we have $\begin{matrix} lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} = eusm 10 M {| X_{t} - Y_{t} | \geq ε} \\ \leq lim_{sup_{s \in [- τ, 0]} | X_{s} - Y_{s} | \to 0} E [{(| X_{t} - Y_{t} |)}^{p}] / ε^{p} = 0, \forall t > 0 \end{matrix}$ by Theorem 2.1. Thus stability in p-th moment implies the stability in measure. □

7 Conclusions

In this paper, we first defined the stability in measure, stability in mean and stability in moment for uncertain delay differential equations. Meanwhile, some theorems on stability in measure, stability in mean and stability in moment were proved, in which the sufficient conditions the uncertain delay differential equation being stable were deduced. At last, we analyzed the relationships among stability in measure, stability in mean as well as stability in moment, and found that stability in p-th moment (0 < p < + ∞) could imply the stability in measure for uncertain delay differential equations. Considering the lack of the analytical solution for most uncertain delay differential equations, we will discuss the numerical method for solving uncertain delay differential equations in future.

Footnotes

Acknowledgment

This work is supported by Natural Science Foundation of Shandong Province (ZR2014GL002) and Higher Educational Science and Technology Program of Shandong Province (12LN15).

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