Stability of high-order uncertain differential equations

Abstract

High-order uncertain differential equations are applied to model differentiable uncertain systems with high-order differentials. In order to describe the influence of the initial value on the solution, this paper proposes two concepts of stability for high-order uncertain differential equation, including stability in measure and stability in mean. Most important of all, some stability theorems are given for a high-order uncertain differential equation. In addition, this paper shows that the given condition is not necessary for a high-order uncertain differential equation being stable via a counterexample. Lastly, this paper will discuss the relationship between stability in measure and stability in mean.

Keywords

High-order uncertain differential equation uncertain process Liu process stability

1 Introduction

For a long time, indeterministic phenomenon was mainly described by probability and fuzzy set theory. However, except for randomness and fuzziness, human uncertainty is another source of indeterminate information, and lots of surveys show that human uncertainty behaves neither like randomness nor fuzziness. In order to deal with the information associated with human uncertainty, an uncertainty theory was founded by Liu [7] using uncertain measure to deal with the belief degree in 2007 and Liu [9] perfected it with presenting product uncertain measure. Uncertain measure, as a set function satisfying the normality, duality, subadditivity and production axioms, is used to indicate the belief degree that an event occurs. Meanwhile, uncertain variable, a basic concept, was proposed by Liu [7]. Uncertain variable, as a measurable function on an uncertainty space, is used to model a quantity with human uncertainty. For describing the uncertain variable, Liu [7] introduced a concept of uncertainty distribution. Besides, Peng and Iwamura [13] gave a sufficient and necessary condition for the uncertainty distribution of an uncertain variable. Liu [9] gave a concept of independence with respect to uncertain variables, based on which the operational law was established by Liu [10] and concepts of expected value, variance and entropy are also proposed to describe an uncertain variable. In addition, the independence of uncertain vectors was discussed by Liu [11].

In order to model the dynamic system involving random factors, stochastic differential equation is proposed that it is a type of differential equations driven by Wiener process. In fuzzy set theory, there exists fuzzy differential equation deal with the dynamic system involving fuzzy factors. Similarly, in uncertainty theory, uncertain process is a sequence of uncertain variables driven by the time or the space. In order to model a dynamic system with human uncertainty, in 2008, Liu [8] gave a concept of uncertain process for modeling the evolution of uncertain phenomena. Then Liu [9] designed a Liu process as a counterpart of standard Wiener process. It is a stationary independent increment uncertain process with normal increments, and its almost all sample paths are Lipschitz continuous. After that, Liu [9] founded an uncertain calculus theory to deal with the integral and differential of an uncertain process with respect to Liu process. Uncertain differential equation is a type of differential equations driven by Liu process, it first proposed by Liu [8], and it aims to describe the evolution of dynation uncertain systems. As the stochastic differential equation [14] and the fuzzy differential equation [1] existence and uniqueness are also the fundamental problems in the uncertain differential equation. Recently, Allahviranloo et al. [2, 3] provided some good ideal to further study the high-order fuzzy differential equation. Chen and Liu [4] provided a sufficient condition for an uncertain differential equation having a unique solution. After a while, Gao [5] gave an existence and unique theorem under weaker conditions. The concept of stability for an uncertain differential equation was proposed by Liu [9] in the sense of uncertain measure. Then Yao et al. [20] gave a sufficient condition for stability. After that, Yao et al. [21] proposed a concept of stability in mean, and gave a sufficient condition. Sheng and Gao [15] proposed exponential stability of uncertain differential equation. Sheng and Wang [16] studied another type of stability in the sense of p-th moment. In 2013, Zhang and Chen [22] proposed a multi-dimensional Liu process and a concept of multi-dimensional uncertain differential equation is proposed by Yao [18] in 2014. After, Ji and Zhou [6] gave existence and uniqueness of solution for a multi-dimensional uncertain differential equation in 2015. Su et al. [17] derived some theorems for multi-dimensional uncertain differential equation being stable.

Recently, Yao [19] proposed a concept of high-order uncertain differential equation to deal with differentiable uncertain systems with high-order differentials, and proved the high-order uncertain differential equation had a unique solution provided that its coefficients satisfy the Lipschitz condition and the linear growth condition. Inspired by these results, this paper will propose some concepts of stability for the high-order uncertain differential equation, and will give a sufficient condition for a high-order uncertain differential equation being stable in measure and in mean. The rest of this paper is organized as follows: Section 2 will introduce some concepts about uncertain variables and Section 3 will introduce high-order uncertain differential equations, respectively. Then Section 4 will introduce the concepts of stability for high-order uncertain differential equation, including stability of in measure and in mean. Following that, this paper will give a sufficient condition for a high-order uncertain differential equation being stable in measure and in mean. Meanwhile, some theorems and remarks of stability will be made, and theirs effectivenessare are illustrated by some examples. Finally, a brief summary is contained in Section 5.

2 Preliminaries

In this section, we will introduce some fundamental concepts and properties concerning uncertain variables, uncertain processes, and uncertain differential equations.

Let Γ be a nonempty set, and ℒ a σ-algebra over Γ. Each element Λ in ℒ is called an event and assigned a number ℳ {Λ} to indicate the belief degree with which we believe Λ will happen. In order to deal with belief degrees rationally, Liu [7] suggested the following three axioms:

Axiom 1. (Normality Axiom) ℳ {Γ} =1 for the universal set Γ;

Axiom 2. (Duality Axiom) ℳ {Λ} + ℳ {Λ^c} =1 for any event Λ;

Axiom 3. (Subadditivity Axiom) For every countable sequence of events Λ₁, Λ₂, ⋯, we have $M {⋃_{i = 1}^{\infty} Λ_{i}} \leq \sum_{i = 1}^{\infty} M {Λ_{i}} .$

Definition 2.1. (Liu [7]) The set function ℳ is called an uncertain measure if it satisfies the normality, duality, and subadditivity axioms.

The triplet (Γ, ℒ, ℳ) is called an uncertainty space. Furthermore, the product uncertain measure on the product σ-algebra ℒ was defined by Liu [9] as follows:

Axiom 4. (Product Axiom) Let (Γ_k, ℒ_k, ℳ_k) be uncertainty spaces for k = 1, 2, ⋯. The product uncertain measure ℳ is an uncertain measure satisfying $M {\prod_{k = 1}^{\infty} Λ_{k}} = ⋀_{k = 1}^{\infty} M_{k} {Λ_{k}}$ where Λ_k are arbitrary events chosen from ℒ_k for k = 1, 2, ⋯, respectively.

Definition 2.2. (Liu [7]) An uncertain variable is a measurable function ξ from an uncertainty space (Γ, ℒ, ℳ) 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 [7]) Suppose ξ is an uncertain variable. Then the uncertainty distribution of ξ is defined by Φ (x) = ℳ { ξ ≤ x } for any real number x .

An uncertainty distribution Φ (x) is said to be regular if its inverse function Φ^-1 (α) exists and is unique for each α ∈ (0, 1). Inverse uncertainty distribution plays an important role in the operations of independent uncertain variables. In the following, the concept of inverse uncertainty distribution will be presented.

The operational law of independent uncertain variables was given by Liu [10] in order to calculate the uncertainty distribution of a strictly increasing or decreasing function of uncertain variables. Before introducing the operational law, the concept of independence of uncertain variables is presented as follows:

Definition 2.4. (Liu [9]) The uncertain variables ξ₁, ξ₂, ⋯ , ξ_n are said to be independent if $M {⋂_{i = 1}^{n} (ξ_{i} \in B_{i})} = ⋀_{i = 1}^{n} M {ξ_{i} \in B_{i}}$ for any Borel sets B₁, B₂, ⋯ , B_n.

For ranking uncertain variables, the concept of expected value was proposed by Liu [7] as follows:

Definition 2.5. (Liu [7]) Let ξ be an uncertain variable. Then the expected value of ξ is defined by $E [ξ] = \int_{0}^{+ \infty} M {ξ \geq x} d x - \int_{- \infty}^{0} M {ξ \leq x} d x$ provided that at least one of the two integrals is finite.

Theorem 2.1. (Liu [7]) Let ξ be an uncertain variable with uncertainty distribution Φ. If the expected value exists, then $E [ξ] = \int_{- \infty}^{+ \infty} x d Φ (x) .$ If the uncertainty distribution Φ is regular, then we also have $E [ξ] = \int_{0}^{1} Φ^{- 1} (α) d α .$

3 High-order uncertain differential equations

An uncertain process is essentially a sequence of uncertain variables indexed by time or space. The study of uncertain process was started by Liu [8] in 2008.

Definition 3.1. (Liu [8]) Let T be an index set and let (Γ, ℒ, ℳ) be an uncertainty space. An uncertain process is a measurable function from T × (Γ, ℒ, ℳ) to the set of real numbers such that {X_t ∈ B} is an event for any Borel set B for each t.

After that, Liu [9] designed a process which is one of the most important uncertain processes, it is named as Liu process thereafter.

Definition 3.2. (Liu [9]) An uncertain process C_t is said to be a Liu process if (i) C₀ = 0 and almost all sample paths are Lipschitz continuous, (ii) C_t has stationary and independent increments, (iii) every increment C_s+t - C_s is a normal uncertain variable with expected value 0 and variance t², whose uncertainty distribution is $Φ (x) = {(1 + exp (\frac{- π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Theorem 3.1. (Yao etal. [20]) Let C_t be a Liu process on an uncertainty space (Γ, ℒ, ℳ). 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, we have $lim_{x \to + \infty} M {K \leq x} = 1$ and

$\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 . \end{matrix}$ (1)

Definition 3.3. (Yao [19]) Let C_t be a Liu process, and f and g are two given functions. Then

$\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ (2) is called an n-order uncertain differential equation. An uncertain process that satisfies (9.1) identically at each time t is called a solution of the high-order uncertain differential equation.

High-order uncertain differential equations are essentially high-order differential equation driven by the Liu process. High-order uncertain differential equations are used to model differentiable uncertain systems with high-order differentials.

Yao showed the high-order uncertain differential equation with an initial value X₀ has a unique solution if the coefficients satisfy the following conditions,

Theorem 3.2. (Yao [19]) The n-order uncertain differential equation (2) has a unique solution if for any (x₁, ⋯ , x_n), $(y_{1}, \dots, y_{n}) \in R^{n}$ and t ≥ 0, the coefficients f (t, x₁, x₂, ⋯ , x_n) and g (t, x₁, x₂, ⋯ , x_n) satisfy the linear growth condition

$\begin{matrix} | f (t, x_{1}, \dots, x_{n}) | + | g (t, x_{1}, \dots, x_{n}) | \\ \leq L (1 + ⋁_{i = 1}^{n} | x_{i} |) \end{matrix}$ (3) and the Lipschitz condition

$\begin{matrix} | f (t, x_{1}, \dots, x_{n}) - f (t, y_{1}, \dots, y_{n}) | \\ + | g (t, x_{1}, \dots, x_{n}) - g (t, y_{1}, \dots, y_{n}) | \\ \leq L \cdot ⋁_{i = 1}^{n} | x_{i} - y_{i} | \end{matrix}$ (4) for some constant L.

For the n-order uncertain differential Equation (2), we denote $X_{1 t} = X_{t}, X_{2 t} = \frac{d X_{t}}{d t}, \dots, X_{nt} = \frac{d^{n - 1} X_{t}}{d t^{n - 1}} .$ Then we have ${\begin{matrix} \frac{d X_{1 t}}{d t} = X_{2 t} \\ \frac{d X_{2 t}}{d t} = X_{3 t} \\ ⋮ \\ \frac{d X_{(n - 1) t}}{d t} = X_{nt} \\ \frac{d X_{nt}}{d t} = f (t, X_{1 t}, X_{2 t}, \dots, X_{nt}) \\ + g (t, X_{1 t}, X_{2 t}, \dots, X_{nt}) \frac{d C_{t}}{d t} . \end{matrix}$ (5)

Denote the vectors $\begin{matrix} X_{t} & = & [X_{1 t}, X_{2 t}, \dots, X_{nt}]^{T} \\ = & {[X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}]}^{T}, \end{matrix}$ $\begin{matrix} f (t, X_{t}) \\ = [X_{2 t}, X_{3 t}, \dots, X_{nt}, f (t, X_{1 t}, X_{2 t}, \dots, X_{nt})]^{T} \end{matrix}$ and $\begin{matrix} g (t, X_{t}) \\ = [0, 0, \dots, g (t, X_{1 t}, X_{2 t}, \dots, X_{nt})]^{T} . \end{matrix}$

Then the n-order uncertain differential equation (2) can be transformed into an n-dimensional uncertain differential equation driven by a Liu process, that is $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t},$ (6) where f (t, X_t), g (t, X_t) are a vector-valued function form $T \times R^{m}$ to $R^{m}$ .

In other words, the equation $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ and the equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ are equivalent.

The uncertain differential equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ is equivalent to the uncertain integral equation $X_{s} = X_{0} + \int_{0}^{s} f (t, X_{t}) d t + \int_{0}^{s} g (t, X_{t}) d C_{t} .$ The n-order uncertain differential equation in the form of $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} \\ = (u_{0 t}, u_{1 t} X_{t}, u_{2 t} \frac{d X_{t}}{d t}, \dots, u_{nt} \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + (v_{0 t}, v_{1 t} X_{t}, v_{2 t} \frac{d X_{t}}{d t}, \dots, v_{nt} \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ is called a linear n-order uncertain differential equation and it can be transformed into a linearn-dimensional uncertain differential equation $d X_{t} = (U_{1 t} X_{t} + U_{2 t}) d t + (V_{1 t} X_{t} + V_{2 t}) d C_{t}$ (7) driven by the Liu process, where $\begin{matrix} U_{1 t} & = & (\begin{matrix} 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 1 \\ u_{1 t} & u_{2 t} & \dots & u_{nt} \end{matrix}), \\ V_{1 t} & = & (\begin{matrix} 0 & 0 & \dots & 0 \\ ⋮ & ⋮ & ⋮ \\ 0 & 0 & \dots & 0 \\ v_{1 t} & v_{2 t} & \dots & v_{nt} \end{matrix}), \\ U_{2 t} & = & {[0, \dots, 0, u_{0 t}]}^{T}, V_{2 t} = {[0, \dots, 0, v_{0 t}]}^{T} . \end{matrix}$

4 Stability of high-order uncertain differential equations

In this section, we will discuss the stability in measure and in mean for higt-order uncertain differential equation. For an n-dimensional vector x = [x₁, x₂, ⋯ , x_m] ^T and an m × n matrix A = [a_ij], we use the infinite normal $| x | = ⋁_{i = 1}^{m} | x_{i} |$ , $| A | = max_{1 \leq i \leq m} \sum_{j = 1}^{n} | a_{ij} | .$

4.1 Stability in measure

In this subsection, we investigate the stability in measure for high-order uncertain differential equation and give a stability theorem.

Definition 4.1. The n-order uncertain differential equation $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ is said to be stable in measure if for any given ɛ ≥ 0, we have $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \leq ɛ} = 1,$ (8) where X_t and Y_t are any two solutions of its equivalent equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ with different initial values X₀ and Y₀.

Example 4.1. The following uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = - \frac{1}{t} exp (- \frac{d X_{t}}{d t}) + exp (- \frac{d X_{t}}{d t}) \cdot \frac{d C_{t}}{d t},$ (9) and its equivalent equation is $\begin{matrix} d X_{t} & = & (\begin{matrix} \frac{d X_{t}}{d t} \\ - \frac{1}{t} exp (- \frac{d X_{t}}{d t}) \end{matrix}) d t \\ + (\begin{matrix} 0 \\ exp (- \frac{d X_{t}}{d t}) \end{matrix}) d C_{t} . \end{matrix}$

It two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & ln (1 + 1 / t^{2} + C_{t}) \cdot X_{0}, \\ Y_{t} & = & ln (1 + 1 / t^{2} + C_{t}) \cdot Y_{0}, \end{matrix}$ restectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | & = & | ln (1 + 1 / t^{2} + C_{t}) \cdot (X_{0} - Y_{0}) | \\ \leq & | ln (1 + 1 / t^{2} + C_{t}) | \cdot | X_{0} - Y_{0} | . \end{matrix}$

As a result, for any given ɛ > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t > 0} | X_{t} - Y_{t} | \leq ɛ} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t > 0} | ln (1 + 1 / t^{2} + C_{t}) | \\ \cdot | X_{0} - Y_{0} | \leq ɛ} \\ = 1 . \end{matrix}$

Thus the 2-order uncertain differential equation is stable in measure.

Example 4.2. The following second order uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = \frac{d X_{t}}{d t} + \frac{d X_{t}}{d t} \cdot \frac{d C_{t}}{d t}$ (10) and its equivalent equation is $d X_{t} = (\begin{matrix} 0 & 1 \\ 0 & 1 \end{matrix}) X_{t} d t + (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) X_{t} d C_{t} .$

It two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & (\begin{matrix} 0 & exp (t) \\ 0 & exp (t + C_{t}) \end{matrix}) \cdot X_{0}, \\ Y_{t} & = & (\begin{matrix} 0 & exp (t) \\ 0 & exp (t + C_{t}) \end{matrix}) \cdot Y_{0}, \end{matrix}$ restectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | \\ = | (\begin{matrix} 0 & exp (t) \\ 0 & exp (t + C_{t}) \end{matrix}) \cdot (X_{0} - Y_{0}) | \\ \leq | (\begin{matrix} 0 & exp (t) \\ 0 & exp (t + C_{t}) \end{matrix}) | \cdot | X_{0} - Y_{0} | \\ = exp (t + C_{t}) \cdot | X_{0} - Y_{0} | . \end{matrix}$

As a result, for any given ɛ > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \leq ɛ} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} exp (t + C_{t}) \\ \cdot | X_{0} - Y_{0} | \leq ɛ} \\ \leq M {C_{t} \geq 0, \forall t \geq 0} \land \\ lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} exp t \cdot | X_{0} - Y_{0} | \leq ɛ} \\ = 1 / 2 . \end{matrix}$

Thus the 2-order uncertain differential equation is not stable in measure.

Theorem 4.1. Suppose the high-order uncertain differential equation $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ is stable in measure if the coefficient functions f (t, x₁, ⋯ , x_n) and g (t, x₁, ⋯ , x_n) satisfy the strong Lipschitz condition

$\begin{matrix} | f (t, x_{1}, \dots, x_{n}) - f (t, y_{1}, \dots, y_{n}) | \\ + | g (t, x_{1}, \dots, x_{n}) - g (t, y_{1}, \dots, y_{n}) | \\ \leq L_{t} \cdot ⋁_{i = 1}^{n} | x_{i} - y_{i} | \end{matrix}$ (11) where L_t is function satisfying $\int_{0}^{+ \infty} L_{t} d t < + \infty .$ (12)

Since the equation $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ and the equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t} .$ are equivalent, so we have the Theorem 4.1 can be transformed into the following Theorem:

Theorem 4.2. Suppose the high-order uncertain differential equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (13) is stable in measure if the coefficient functions f (t, X_t) and g (t, X_t) satisfy the strong Lipschitz condition

$\begin{matrix} | f (t, x) - f (t, y) | + | g (t, x) - g (t, y) | \\ \leq L_{t} | x - y |, \forall x, y \in R^{m}, t \geq 0 \end{matrix}$ (14) where $| x - y | = ⋁_{i = 1}^{n} | x_{i} - y_{i} |$ and L_t are functions satisfying $\int_{0}^{+ \infty} L_{t} d t < + \infty .$ (15)

Proof. Assume that X_t and Y_t are two solutions of the n-order uncertain differential Equation (13) with two different initial values X₀ and Y₀, respectively, i.e., $\begin{matrix} X_{t} & = & X_{0} + \int_{0}^{t} f (s, X_{s}) d s + \int_{0}^{t} g (s, X_{s}) d C_{s}, \\ Y_{t} & = & Y_{0} + \int_{0}^{t} f (s, Y_{s}) d s + \int_{0}^{t} g (s, Y_{s}) d C_{s} . \end{matrix}$

Then we have $\begin{matrix} X_{t} - Y_{t} \\ = (X_{0} - Y_{0}) + \int_{0}^{t} (f (s, X_{s}) - f (s, Y_{s})) d s \\ + \int_{0}^{t} (g (s, X_{s}) - g (s, Y_{s})) d C_{s} . \end{matrix}$

According to the strong Lipschitz condition, we can obtain $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | + | \int_{0}^{t} f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) d s | \\ + | \int_{0}^{t} | g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) d s \\ + \int_{0}^{t} | g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) | K (γ) d s \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} L_{s} | X_{s} (γ) - Y_{s} γ) | d s \\ + \int_{0}^{t} L_{s} K (γ) | X_{s} (γ) - Y_{s} (γ) | d s \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} (1 + K (γ)) L_{s} | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$

where K (γ) is the Lipschitz constants of C_t (γ). By the Gronwall’s inequality, for any t ≥ 0, we can also obtain $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{t} (1 + K (γ)) L_{s} d s) \\ \leq | X_{0} - Y_{0} | exp ((1 + K (γ)) \int_{0}^{+ \infty} L_{s} d s) . \end{matrix}$

So we have

$\begin{matrix} sup_{t \geq 0} | X_{t} - Y_{t} | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{s} d s) \\ \cdot exp (K \int_{0}^{+ \infty} L_{s} d s) \end{matrix}$ (16) almost surely, where K is a nonnegative uncertain variable such that $\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3} σ}))}^{- 1} - 1 . \end{matrix}$ by Theorem 3.1 .

Then for any given ε ≥ 0, there exists a real number T = T (ε) such that $M {γ \in Γ | K (γ) \leq T} \geq 1 - ε .$ We take $δ = exp (- (1 + T) \int_{0}^{+ \infty} L_{s} d s) ɛ .$

Then we have |X_t (γ) - Y_t (γ) | ≤ ɛ, ∀t ≥ 0 provided that |X₀ - Y₀| ≤ δ and K (γ) ≤ T. So if |X₀ - Y₀| ≤ δ we have $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \leq ɛ} = 1,$

Hence, it follows from the definition of stability in measure that the high-order uncertain differential equation is stable in measure under the strong Lipschitz condition. The theorem is proved.

Example 4.3. The following 2-order uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = - exp (- t^{2}) \frac{d X_{t}}{d t} + exp (- t) \frac{d X_{t}}{d t} \cdot \frac{d C_{t}}{d t} .$ (17)

Its equivalent equation is $\begin{matrix} d X_{t} & = & (\begin{matrix} 0 & exp (- t^{2}) \\ 0 & - exp (- t^{2}) \end{matrix}) X_{t} d t \\ + (\begin{matrix} 0 & 0 \\ 0 & exp (- t) \end{matrix}) X_{t} d C_{t} . \end{matrix}$

Note that the function f (t, x) = exp(- t²) · x satisfies $| f (t, x) - f (t, y) | \leq exp (- t^{2}) \cdot | x - y |$ and the function g (t, x) = exp(- t) · x satisfies $| g (t, x) - g (t, y) | \leq exp (- t) \cdot | x - y | .$ Since the function L_t = exp(- t²) + exp(- t) satisfies $\int_{0}^{+ \infty} L_{t} d t < + \infty,$ the 2-order uncertain differential equation is stable in measure.

Remark 4.1. Theorems 4.1, 4.2 give the sufficient condition but not the necessary condition for high-order uncertain differential equation being stable in measure.

4.2 Stability in mean

In this subsection, we investigate the stability in mean for high-order uncertain differential equation and give some stability theorems.

Definition 4.2. The high-order uncertain differential equation $\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ is said to be stable in mean if for any given ɛ ≥ 0, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} | \leq ɛ] = 1,$ (18) where X_t and Y_t are any two solutions of its equivalent equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ with different initial values X₀ and Y₀.

Example 4.4. The following uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = - \frac{d X_{t}}{d t} + - \frac{d X_{t}}{d t} \cdot \frac{d C_{t}}{d t} .$ (19)

It two solutions with different initial values X₀ and Y₀ are $\begin{matrix} X_{t} & = & X_{0} \int_{0}^{t} exp (- s + C_{s}) d s, \\ Y_{0} & = & Y_{0} \int_{0}^{t} exp (- s + C_{s}) d s, \end{matrix}$ restectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | \\ = | \int_{0}^{t} exp (- s + C_{s}) d s \cdot (X_{0} - Y_{0}) | \\ \leq | \int_{0}^{t} exp (- s + C_{s}) d s | \cdot | X_{0} - Y_{0} | . \end{matrix}$

As a result, for any given ɛ > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t > 0} | X_{t} - Y_{t} | \leq ɛ] \\ \leq E [sup_{t \geq 0} | \int_{0}^{t} exp (- s + C_{s}) d s | \cdot | X_{0} - Y_{0} | \leq ɛ] \\ = 1 . \end{matrix}$

Thus the 2-order uncertain differential equation is stable in mean.

Theorem 4.3. Suppose the n-order uncertain differential equation

$\begin{matrix} \frac{d^{n} X_{t}}{d t^{n}} & = & f (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \\ + g (t, X_{t}, \frac{d X_{t}}{d t}, \dots, \frac{d^{n - 1} X_{t}}{d t^{n - 1}}) \frac{d C_{t}}{d t} \end{matrix}$ (20) and its equivalent equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (21) is stable in mean if the coefficient functions f (t, X_t) and g (t, X_t) satisfy the strong Lipschitz condition

$\begin{matrix} | f (t, x) - f (t, y) | \leq L_{1 t} | x - y |, \\ | g (t, x) - g (t, y) | \leq L_{2 t} | x - y |, \\ \forall x, y \in R^{m}, t \geq 0 \end{matrix}$ (22) where L_1t and L_2t are two functions satisfying $\int_{0}^{+ \infty} L_{1 t} d t < + \infty, \int_{0}^{+ \infty} L_{2 t} d t < \frac{π}{\sqrt{3}} .$ (23)

Proof. Assume that X_t and Y_t are two solutions of the n-order uncertain differential Equation (27) with two different initial values X₀ and Y₀, respectively, i.e., $\begin{matrix} X_{t} & = & X_{0} + \int_{0}^{t} f (s, X_{s}) d s + \int_{0}^{t} g (s, X_{s}) d C_{s}, \\ Y_{t} & = & Y_{0} + \int_{0}^{t} f (s, Y_{s}) d s + \int_{0}^{t} g (s, Y_{s}) d C_{s} . \end{matrix}$

Then we have $\begin{matrix} X_{t} - Y_{t} \\ = (X_{0} - Y_{0}) + \int_{0}^{t} (f (s, X_{s}) - f (s, Y_{s})) d s \\ + \int_{0}^{t} (g (s, X_{s}) - g (s, Y_{s})) d C_{s} \\ = (X_{0} - Y_{0}) + \int_{0}^{t} (f (s, X_{s}) - f (s, Y_{s})) d s \\ + \int_{0}^{t} (g (s, X_{s}) - g (s, Y_{s})) d C_{s} . \end{matrix}$

According to the strong Lipschitz condition, we can obtain $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | + | \int_{0}^{t} f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) d s | \\ + | \int_{0}^{t} g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) | K (γ) d s \\ \leq | X_{0} - Y_{0} | + \int_{0}^{t} L_{s} | X_{s} (γ) - Y_{s} γ) | d s \\ + \int_{0}^{t} L_{s} K (γ) | X_{s} (γ) - Y_{s} (γ) | d s \\ \leq | X_{0} - Y_{0} | \\ + \int_{0}^{t} (1 + K (γ)) L_{s} | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$

So we have

$\begin{matrix} sup_{t \geq 0} | X_{t} - Y_{t} | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{s} d s) \\ \cdot exp (K \int_{0}^{+ \infty} L_{s} d s) \end{matrix}$ (24) almost surely, where K is a nonnegative uncertain variable, we have $\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3} σ}))}^{- 1} - 1 . \end{matrix}$ by Theorem 3.1.

Taking expected value on both sides of (24), we have $\begin{matrix} E [sup_{t \geq 0} | X_{t} - Y_{t} |] \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} d s) \\ \cdot E [exp (K \int_{0}^{+ \infty} L_{2 s} d s)] . \end{matrix}$

Since $\int_{0}^{+ \infty} L_{1 s} d s < + \infty,$ we have $E [exp (\int_{0}^{+ \infty} L_{1 s} d s)] < + \infty,$ and if $\int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3}},$ then we have $\begin{matrix} E [exp (K \int_{0}^{+ \infty} L_{2 s} d s)] \\ = \int_{0}^{+ \infty} M {exp (K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ = \int_{0}^{1} M {exp (K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ + \int_{1}^{+ \infty} M {exp (K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ \leq 1 + \int_{1}^{+ \infty} M {exp (\int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ = 1 + \int_{0}^{+ \infty} L_{2 s} d s \\ \cdot \int_{0}^{+ \infty} exp (y \int_{0}^{+ \infty} L_{2 s} d s) M {K \geq y} d y \\ = 1 + \int_{0}^{+ \infty} L_{2 s} d s \int_{0}^{+ \infty} exp (y \int_{0}^{+ \infty} L_{2 s} d s) \\ \cdot {1 - (2 {(1 + exp (\frac{- π x}{\sqrt{3} σ}))}^{- 1} - 1)} d y \\ = 1 + 2 (\int_{0}^{+ \infty} L_{2 s} d s) \cdot \int_{0}^{+ \infty} exp (y \int_{0}^{+ \infty} L_{2 s} d s) \\ \cdot (1 + exp {(\frac{- π x}{\sqrt{3} σ})}^{- 1}) d y \\ = 1 + 2 \int_{1}^{+ \infty} {1 + x^{π / (\sqrt{3} \int_{0}^{+ \infty} L_{2 s} d s)}}^{- 1} d x \\ < + \infty . \end{matrix}$

So we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |] \\ \leq lim_{| X_{0} - Y_{0} | \to 0} | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} d s) \\ \cdot E [exp (K \int_{0}^{+ \infty} L_{2 s} d s)] \\ = 0 . \end{matrix}$

Hence, it follows from the definition of stability in mean that the high-order uncertain differential equation is stable in mean under the strong Lipschitz condition. The theorem is proved.

Example 4.5. The following 2-order uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = - exp (- t^{2}) \frac{d X_{t}}{d t} + exp (- t) \frac{d X_{t}}{d t} \cdot \frac{d C_{t}}{d t} .$ (25)

Since the function L_1t = exp(- t²) satisfies $\int_{0}^{+ \infty} L_{1 t} d t < + \infty,$ and L_2t = exp(- t) satisfies $\int_{0}^{+ \infty} L_{2 t} d t = 1 < \frac{π}{\sqrt{3}},$ the 2-order uncertain differential equation is stable in mean.

Remark 4.2 Theorem 4.3 gives the sufficient condition but not the necessary condition for high-order uncertain differential equation being stable in mean.

4.3 Relationship of stability in measure and stability in mean

In this section, the relationship between stability in measure and stability in mean for a high-order uncertain differential equation is discussed.

Theorem 4.4. If an uncertain differential equation is stable in mean, then it is stable in measure.

Proof. It follows from the definition of stability in mean that for two solutions X_t and Y_t with different initial values X₀ and Y₀, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |] = 0, \forall t \geq 0 . \end{matrix}$

Then for any given real number ɛ ≥ 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \geq ɛ} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} \frac{E [{sup}_{t \geq 0} | X_{t} - Y_{t} |]}{ɛ} \\ = 0, \end{matrix}$ ∀t ≥ 0 by Markov inequality. Thus, for a high-order uncertain differential equation is stable in mean implies stable in measure.

Remark 4.3. For a high-order uncertain differential equation, generally, stability in measure does not imply stability in mean.

Example 4.6. The following 2-order uncertain differential equation $\frac{d^{2} X_{t}}{d t^{2}} = - exp (- t^{2}) \frac{d X_{t}}{d t} + exp (- 2 t) \frac{d X_{t}}{d t} \cdot \frac{d C_{t}}{d t} .$ (26)

Its equivalent equation is $\begin{matrix} d X_{t} & = & (\begin{matrix} 0 & exp (- t^{2}) \\ 0 & - exp (- t^{2}) \end{matrix}) X_{t} d t \\ + (\begin{matrix} 0 & 0 \\ 0 & exp (- 2 t) \end{matrix}) X_{t} d C_{t} . \end{matrix}$

Note that the function f (t, x) = exp(- t²) · x satisfies $| f (t, x) - f (t, y) | \leq exp (- t^{2}) \cdot | x - y |$ and the function g (t, x) = exp(-2t) · x satisfies $| g (t, x) - g (t, y) | \leq exp (- 2 t) \cdot | x - y | .$

Since the function L_t = exp(- t²) + exp(-2t) satisfies $\int_{0}^{+ \infty} L_{t} d t < + \infty,$ the 2-order uncertain differential equation is stable in measure. But the function L_1t = exp(- t²) satisfies $\int_{0}^{+ \infty} L_{1 t} d t < + \infty,$ and L_2t = exp(-2t) satisfies $\int_{0}^{+ \infty} L_{2 t} d t = 2 > \frac{π}{\sqrt{3}},$ the 2-order uncertain differential equation is not stable in mean.

5 Conclusions

High-order uncertain differential equations describe the dynamic uncertain systems involving high-order differentials. It is essentially a system of uncertain differential equations. This paper proposed some concepts of stability for a high-order uncertain differential equation. Some theorems on stability in measure and in mean were proved, in which the sufficient conditions the high-order uncertain differential equation being stable in measure and in mean were provided. In addition, the relationship between stability in measure and stability in mean were discussed about the high-order uncertain differential equation.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grants Nos. 61563050, 61462086) and Doctoral Fund of Xinjiang University (No. BS150206).

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