Stability in p -th moment of multi-dimensional uncertain differential equation

Abstract

Uncertain differential equation plays an important role in dealing with dynamical systems with uncertainty. Multi-dimensional uncertain differential equation is a type of differential equation driven by multi-dimensional Liu processes. Stability analysis of a multi-dimensional means insensitivity of the state of a system to small changes in the initial state. This paper focuses on the stability in p-th moment for multi-dimensional uncertain differential equation. The concept of stability in p-th moment for multi-dimensional uncertain differential equation is presented. Some stability theorems for the solution of multi-dimensional uncertain differential equation are given, in which some sufficient conditions for a multi-dimensional uncertain differential equation being stable in p-th moment and a sufficient and necessary condition for a linear multi-dimensional uncertain differential equation being stable in p-th moment are provided. In addition, this paper discusses the relationships among stability in p-th moment, stability in measure and stability in mean.

Keywords

Uncertainty theory multi-dimensional uncertain differential equation moment stability

1 Introduction

In the framework of probability theory, Wiener [23] defined a stochastic process in 1923, which is named as Wiener process thereafter. For handling dynamic stochastic systems, Ito [4, 5] founded stochastic calculus in 1940s to deal with stochastic differential equation with respect to Wiener process. In the 70s and 80s of last century, stochastic differential equation and stochastic analysis developed rapidly and were used widely in many aspects. Except for random phenomena in reality, human’s uncertainty associated with belief degree is another different type of indeterminate phenomenon. For modeling human uncertainty associated with belief degrees, uncertainty theory was founded by Liu [9] in 2007 based on normality, duality, subadditivity axioms. And Liu [11] refined uncertainty theory by presenting product axiom which is completely different from the one in probability theory. Uncertainty theory becomes an almost completely theoretical system.

In order to describe the evolution of an uncertain phenomenon, an uncertain process was proposed by Liu [10] as a sequence of uncertain variables indexed by time. After that, Liu [11] presented a Liu process which is also a type of stationary independent increment process, but the increments are normal uncertain variables instead of random variables, and in addition, almost all sample paths of Liu process are Lipschitz continuity instead of non-Lipschitz continuity. Meanwhile, Liu [11] founded uncertain calculus to handle the integral and the differential of an uncertain process with respect to Liu process, Chen and Ralescu [2] proposed an uncertain integral with respect to general Liu process. Uncertain integral was extended from single Liu process to multiple ones by Liu and Yao [14].

Uncertain differential equation was first proposed by Liu [10] in 2008. Then Chen and Liu [1] proved that the existence and uniqueness theorem of solution of uncertain differential equations if its coefficients satisfy the linear growth condition and Lipschitz continuous condition. After that, Gao [3] gave an existence and uniqueness theorem under local linear growth condition and local Lipschitz continuous condition. In addition, stability of uncertain differential equation has recently received a lot of attentions. Stability of a differential equation means that a perturbation on the initial value will not result in an influential shift. The concept of stability in measure of uncertain differential equations was presented by Liu [11], and Yao et al. [27] proved some stability theorems of uncertain differential equation. Following that, Yao et al . [28], Sheng and Wang [19], Sheng and Gao [18], Liu et al. [16] and Yang et al . [30] discussed stability in mean, stability in p-th moment, exponential stability, almost sure stability and stability in inverse distribution of uncertain differential equations, respectively. In 2014, Yao [29] discussed multi-dimensional uncertain differential equations consider dynamic system being effected by various kinds of uncertain factors, and proved existence and uniqueness of its solution. Zhang and Chen [31] showed that multi-dimensional Liu processes is a stationary independent increment multi-dimensional uncertain process, and its almost all sample paths are Lipschitz continuous. Then Su et al . [22] proposed the stability of the solution for such differential equations in the sense of uncertain measure. Subsequently, Sheng and Shi [21] investigated the stability in mean of the solution of multi-dimensional uncertain differential equations.

Extending the previous work on stability of multi-dimensional uncertain differential equation, this paper will develop a concept of stability in p-th moment of multi-dimensional uncertain differential equation and give a sufficient condition for multi-dimensional uncertain differential equation having stability in p-th moment. The relationship among the stability in p-th moment, the stability in mean and the stability in measure for multi-dimensional uncertain differential equation is also discussed. The rest of the paper is organized as follows. In Section 2, we will briefly review some basic concepts and properties of uncertain variable, uncertain process and uncertain differential equation. Then we present the concept of stability in p-th moment for multi-dimensional uncertain differential equation. In Section 3, some stability theorems of a multi-dimensional differential equation are given. A sufficient and necessary condition for a linear multi-dimensional uncertain differential equation being stable in p-th moment is proved. The relationship between the stability in measure and the stability in p-th moment for uncertain differential equation is discussed in Section 4. At last, some conclusions are given in Section 5.

2 Preliminary

2.1 Uncertainty variable

In this part, we review some preliminary concepts and properties in uncertainty variables.

Let $L$ be a σ-algebra on a nonempty set 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 [9] 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 1. (Liu [9]) 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, Liu [11] defined the product uncertain measure on the product σ-algebra $L$ as follows:

Axiom 4. (Product Axiom) Let $(Γ_{k}, L_{k}, M_{k})$ be uncertainty spaces for k = 1, 2, ⋯ Then 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 arbitrarily chosen events from $L_{k}$ for k = 1, 2, ⋯, respectively.

Definition 2. (Liu [9]) 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 ${ξ \in B} = {γ \in Γ | ξ (γ) \in B}$ is an event.

An uncertain variable is essentially a measurable function from an uncertainty space to the set of real numbers. In order to describe an uncertain variable, a concept of uncertainty distribution is defined as follows.

Definition 3. (Liu [9]) The uncertainty distribution Φ of an uncertain variable ξ is defined by $Φ (x) = M {ξ \leq x}$ for any $x \in R .$

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). The inverse function Φ^-1 (α) is called the inverse uncertainty distribution of ξ. Inverse uncertainty distribution plays an important role in the operations of independent uncertain variables.

Theorem 1. (Liu [12]) A function $Φ^{- 1} (α) : (0, 1) \to R$ is an inverse uncertainty distribution if and only if it is a continuous and strictly increasing function with respect to α.

The operational law of independent uncertain variables was given by Liu [9] 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 4. (Liu [11]) The uncertainty 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.

Theorem 2. (Liu [12]) Let ξ₁, ξ₂, ⋯ , ξ_n be independent uncertain variables with regular uncertainty distributions Φ₁, Φ₂, ⋯ , Φ_n, respectively. If f is a strictly increasing function for ξ₁, ξ₂, ⋯ , ξ_m and f is a strictly decreasing function for ξ_m+1, ξ_m+2, ⋯ , ξ_n, then f (ξ₁, ξ₂, ⋯ , ξ_n) is an uncertain variable with inverse uncertainty distribution $\begin{matrix} Ψ^{- 1} (α) = f (Φ_{1}^{- 1} (α), Φ_{2}^{- 1} (α), \dots, Φ_{m}^{- 1} (α), \\ Φ_{m + 1}^{- 1} (1 - α), Φ_{m + 2}^{- 1} (1 - α), \dots, Φ_{n}^{- 1} (1 - α)) . \end{matrix}$

For ranking uncertain variables, the concept of expected value was proposed by Liu [9] as follows:

Definition 5. (Liu [9]) The expected value of an uncertain variable 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 exists.

For an uncertain variable ξ with a regular uncertainty distribution Φ, we have $\begin{matrix} E [ξ] = \int_{0}^{1} Φ^{- 1} (α) d α . \end{matrix}$ The expected value of a normal uncertain variable ξ is E [ξ] = e, and the expected value of a lognormal uncertain variable exp(ξ) is $E [exp (ξ)] = {\begin{matrix} \sqrt{3} σ exp (e) csc (\sqrt{3} σ), & if σ \leq π / \sqrt{3} \\ + \infty, & otherwise . \end{matrix}$ (1)

2.2 Uncertain calculus

An uncertain process is essentially a sequence of uncertain variables indexed by time. The study of uncertain process was started by Liu [10] in 2008.

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

Definition 6. (Liu [11]) 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 $Φ_{t} (x) = {(1 + exp (- \frac{π x}{\sqrt{3} t}))}^{- 1}, x \in R .$

Note that the Lipschitz constant of the sample path C_t (γ) is a function of γ, so all the Lipschitz constants could be regarded as the values that an uncertain variable may take. The following theorem gives a lower bound of the uncertainty distribution of such an uncertain variable.

Let X_t be an uncertain process and C_t be Liu process. For any partition of closed interval [a, b] with a = t₁ < t₂ < ⋯ < t_k+1 = b, the mesh is written as $Δ = max_{1 \leq i \leq k} | t_{i + 1} - t_{i} | .$ 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. In this case, the uncertain process X_t is said to be integrable.

For example, an integrable function f (t) is an integrable uncertain process, and the uncertain integral $\int_{0}^{s} f (t) d C_{t} \sim N (0, \int_{0}^{s} | f (t) | d t)$ (2) is a normal uncertain variable.

Theorem 3. (Yao et al. [27]) 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 $lim_{x \to + \infty} M {K \leq x} = 1$ (3) and $M {γ \in Γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3} σ}))}^{- 1} - 1 .$ (4)

Definition 7. (Liu [15]) Uncertain processes X_1t, X_2t, ⋯, X_nt are said to be independent if for any positive integer k and any times t₁, t₂, ⋯ , t_k, the uncertain vectors $ξ_{i} = (X_{{it}_{1}}, X_{{it}_{2}}, \dots, X_{{it}_{k}}), i = 1, 2, \dots, n$ are independent.

Definition 8. (Zhang and Chen [31]) Let C_it, i = 1, 2, ⋯ , n be independent Liu processes on an uncertainty space $(Γ, L, M)$ . Then C_t = (C_1t, C_2t, ⋯ , C_nt) ^T is called an n-dimensional Liu process on the uncertainty space $(Γ, L, M)$ .

Zhang and Chen [31] showed that C_t is a stationary independent increment multi-dimensional uncertain process, and its almost all sample paths are Lipschitz continuous.

Definition 9. (Yao [29]) Let C_t = (C_1t, C_2t, ⋯, C_nt) ^T be an n-dimensional Liu processes, and let X_t = [X_ijk] be an m × n uncertain matrix process whose elements X_ijk are integrable uncertain processes. Then the uncertain integral of X_t with respect to the n-dimensional Liu process C_t is defined by $\begin{matrix} \int_{a}^{b} X_{t} d C_{t} = [\sum_{j = 1}^{n} \int_{a}^{b} X_{1 jt} d C_{jt}, \\ \sum_{j = 1}^{n} \int_{a}^{b} X_{2 jt} d C_{jt}, \dots, \sum_{j = 1}^{n} \int_{a}^{b} X_{mjt} d C_{jt}]^{T} . \end{matrix}$

Definition 10. (Yao [29]) Let C_t be an n-dimensional Liu processes. Assume that f (t, x) is a vector valued function from $T \times R^{m}$ to $R^{m}$ , and g (t, x) is a matrix-valued function from $T \times R^{m}$ to m × n matrices. Then $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ (5) is called an m-dimensional uncertain differential equation with respect to C_t. The solution is an n-dimensional uncertain process such that Equation (5) holds for each t.

Essentially, the multi-dimensional uncertain differential equation is a type of differential equation driven by multiple Liu processes.

Definition 11. (Su et al. [22]) A multi-dimensional uncertain differential equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ is called stable in measure if for any given real number ε ≥ 0, we have $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \leq ε} = 1,$ (6) where X_t and Y_t are any two solutions with different initial values X₀ and Y₀.

Definition 12. (Sheng and Shi [21]) A multi-dimensional uncertain differential equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{t}$ is called stable in mean if $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |] = 0,$ (7) where X_t and Y_t are any two solutions with different initial values X₀ and Y₀.

3 Stability in p-th moment

In this section, a concept of stability in p-th moment for a multi-dimensional uncertain differential equation, give the conditions stability in p-th moment for this type uncertain differential equation. First we explain the meaning of some symbols. For an m-dimensional vector x = (x₁, x₂, ⋯ , x_m) and an m × n matrix B = [b_ij], we use the infinite normal

$| x | = max_{1 \leq i \leq m} | x_{i} |, | A | = max_{1 \leq i \leq m} \sum_{j = 1}^{n} | a_{ij} | .$

Definition 13. A multi-dimensional uncertain differential equation $d X_{t} = f (t, X_{t}) d t + g (t, X_{t}) d C_{it}$ (8) is said to be stable in p-th moment if for any solutions X_t and Y_t with different initial values X₀ and Y₀, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] = 0, \forall 0 < p < \infty .$ (9)

In particular, when p=1, if $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |] = 0,$ then the multi-dimensional uncertain differential equation (10) is said to be stable in mean. (Sheng and Shi [21])

Example 1. The following 2-dimensional uncertain differential equation $d X_{t} = (\begin{matrix} 0 1 \\ 0 1 \end{matrix}) X_{t} d t + (\begin{matrix} 0 0 \\ 0 1 \end{matrix}) X_{t} d C_{t} .$ It two solutions with different initial values X₀ and Y₀ are $X_{t} = (\begin{matrix} 0 exp (t + C_{t}) \\ 0 exp (t + C_{t}) \end{matrix}) X_{0}$ and $Y_{t} = (\begin{matrix} 0 0 \\ 0 exp (t + C_{t}) \end{matrix}) Y_{0},$ restectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | = | (\begin{matrix} 0 exp (t + C_{t}) \\ 0 0 \end{matrix}) (X_{0} - Y_{0}) | \\ \leq & | (\begin{matrix} 0 0 \\ 0 exp (t + C_{t}) \end{matrix}) | | X_{0} - Y_{0} | \\ = & exp (t + C_{t}) | X_{0} - Y_{0} | . \end{matrix}$ As a result, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E {sup_{t \geq 0} | X_{t} - {Y_{t}}^{p} |} \\ \leq & lim_{| X_{0} - Y_{0} | \to 0} E {sup_{t \geq 0} exp (t + C_{t}) | X_{0} - Y_{0} |^{p}} \\ = & lim_{| X_{0} - Y_{0} | \to 0} E {sup_{t \geq 0} exp t | X_{0} - Y_{0} |^{p}} = 0 . \end{matrix}$

Thus the 2-dimensional uncertain differential equation is stable in p-th moment.

Example 2. The following 2-dimensional uncertain differential equation $d X_{t} = (\begin{matrix} μ_{1 t} \\ μ_{2 t} \end{matrix}) d t + (\begin{matrix} σ_{1 t} 0 \\ 0 σ_{2 t} \end{matrix}) (\begin{matrix} d C_{1 t} \\ d C_{2 t} \end{matrix}) .$ It two solutions with different initial values X₀ and Y₀ are $X_{t} = X_{0} + (\begin{matrix} \int_{0}^{t} μ_{1 s} d s + \int_{0}^{t} σ_{1 s} d C_{1 t} \\ \int_{0}^{t} μ_{2 s} d s + \int_{0}^{t} σ_{2 s} d C_{2 s} \end{matrix})$ and $Y_{t} = Y_{0} + (\begin{matrix} \int_{0}^{t} μ_{1 s} d s + \int_{0}^{t} σ_{1 s} d C_{1 t} \\ \int_{0}^{t} μ_{2 s} d s + \int_{0}^{t} σ_{2 s} d C_{2 s} \end{matrix}),$ restectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | = | X_{0} - Y_{0} | \end{matrix}$ we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{0} - Y_{0} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [| X_{0} - Y_{0} |^{p}] = 0 . \end{matrix}$

Thus the 2-dimensional uncertain differential equation is stable in p-th moment.

Example 3. The following 2-dimensional uncertain differential equation $d X_{t} = - X_{t} d t + (\begin{matrix} σ_{1} 0 \\ 0 σ_{2} \end{matrix}) (\begin{matrix} d C_{1 t} \\ d C_{2 t} \end{matrix}) .$ It two solutions with different initial values X₀ and Y₀ are $X_{t} = exp (- t) X_{0} + (\begin{matrix} \int_{0}^{t} σ_{1} exp (s - t) d C_{1 s} \\ \int_{0}^{t} σ_{2} exp (s - t) d C_{2 s} \end{matrix})$ and $Y_{t} = exp (- t) Y_{0} + (\begin{matrix} \int_{0}^{t} σ_{1} exp (s - t) d C_{1 s} \\ \int_{0}^{t} σ_{2} exp (s - t) d C_{2 s} \end{matrix}),$ restectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | = exp (- t) | X_{0} - Y_{0} | \end{matrix}$ we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} exp (- pt) | X_{0} - Y_{0} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [| X_{0} - Y_{0} |^{p}] = 0 . \end{matrix}$

Thus the 2-dimensional uncertain differential equation is stable in p-th moment.

Example 4. The following m-dimensional uncertain differential equation $d X_{t} = U_{t} d t + V_{t} d C_{t}$ where U_t is an m-dimensional integrable uncertain process, V_t is an m × n integrable uncertain matrix process, and C_t is an n-dimensional Liu process. It two solutions with different initial values X₀ and Y₀ are $X_{t} = X_{0} + \int_{0}^{t} U_{s} d s + \int_{0}^{t} V_{s} d C_{t}$ and $Y_{t} = Y_{0} + \int_{0}^{t} U_{s} d s + \int_{0}^{t} V_{s} d C_{t},$ restectively. Then we have $\begin{matrix} sup_{t \geq 0} | X_{t} - Y_{t} | = sup_{t \geq 0} | X_{0} - Y_{0} | = | X_{0} - Y_{0} | \end{matrix}$ As a result, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{0} - Y_{0} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [| X_{0} - Y_{0} |^{p}] = 0 \end{matrix}$ Thus the m-dimensional uncertain differential equation is stable in p-th moment.

4 Stability theorem

In this part, we provide a sufficient condition for a multi-dimensional differential equation being stable in p-th moment. Besides, we give an example to show that the condition is not necessary.

Theorem 4. The multi-dimensional uncertain differential equation $d X_{t} = f (t, X_{t}) + g (t, X_{t}) d C_{t}$ (10) is stable in p-th moment if the coefficient functions f (t, x) and g (t, x) satisfy the strong Lipschitz condition $| f (t, x) - f (t, y) | \leq L_{1 t} | x - y |, \forall x, y \in R^{m}, t \geq 0$ (11) and $| g (t, x) - g (t, y) | \leq L_{2 t} | x - y |, \forall x, y \in R^{m}, t \geq 0 .$ (12) where L_1t and L_2t are two functions satisfying $sup_{t \geq 0} \int_{0}^{+ \infty} L_{1 s} d s < + \infty and \int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3} p} .$ (13)

Proof. Assume that X_t and Y_t are two solutions of the multi-dimensional uncertain differential equation (18) 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 $\begin{matrix} X_{t} (γ) - Y_{t} (γ) \\ = & (X_{0} - Y_{0}) + \int_{0}^{t} (f (s, X_{s} (γ)) - f (s, Y_{s} (γ))) d s \\ + \int_{0}^{t} (g (s, X_{s} (γ)) - g (s, Y_{s} (γ))) d C_{s} (γ) \end{matrix}$

According to the strong Lipschitz condition, we can obtain $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq & | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) | d | C_{s} (γ) | \\ \leq & | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) | K (γ) d s \\ \leq & | X_{0} - Y_{0} | + \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \int_{0}^{t} | g (s, X_{s} (γ)) - g (s, Y_{s} (γ)) | K (γ) d s \\ \leq & | X_{0} - Y_{0} | + \int_{0}^{t} L_{1 s} | X_{s} (γ) - Y_{(} γ) | d s \\ + \int_{0}^{t} L_{2 s} K (γ) | X_{s} (γ) - Y_{s} (γ) | d s \\ \leq & | X_{0} - Y_{0} | \\ + \int_{0}^{t} (L_{1 s} + L_{2 s} K (γ)) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$ where $K (γ) = {max}_{i = 1}^{n} K_{i} (γ)$ and K_i (γ) are the Lipschitz constants of C_it (γ), i = 1, 2, ⋯ , n, respectively, and K_i (γ), i = 1, 2, ⋯ , n are independent. By the Gronwall’s inequality, we can obtain $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq & | X_{0} - Y_{0} | exp (\int_{0}^{t} L_{1 s} d s) \\ exp (K (γ) \int_{0}^{t} L_{2 s} d s) \\ \leq & | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} d s) \\ exp (K (γ) \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ for any t ≥ 0. So we have $\begin{matrix} sup_{t \geq 0} | X_{t} - Y_{t} | \leq | X_{0} - Y_{0} | \\ exp (\int_{0}^{+ \infty} L_{1 s} d s) exp (K \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ (14) almost surely, where K is a nonnegative uncertain variable. Since {K (γ) ≤ x} and ${⋂_{i = 1}^{n} K_{i} (γ) \leq x}$ is equivalent, then we have $\begin{matrix} M {γ \in Γ | K (γ) \leq x} \\ \geq & 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 \end{matrix}$ by Theorem 3 and independence of K_i (γ), i = 1, 2, ⋯ , n. Taking p-th moment on both sides for equation (14), we have $\begin{matrix} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \leq | X_{0} - Y_{0} |^{p} \\ exp (p \int_{0}^{+ \infty} L_{1 s} d s) E [p exp (K \int_{0}^{+ \infty} L_{2 s} d s)] . \end{matrix}$ Since $sup_{t \geq 0} \int_{0}^{+ \infty} L_{1 s} d s < + \infty,$ we have $E [exp (p \int_{0}^{+ \infty} L_{1 s} d s)] < + \infty,$ and since $\int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3} p},$ it follows from the definition of expected value that we have $\begin{matrix} E [exp (p K \int_{0}^{+ \infty} L_{2 s} d s)] \\ = & \int_{0}^{+ \infty} M {exp (p K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ = & \int_{0}^{1} M {exp (p K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ + \int_{1}^{+ \infty} M {exp (p K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ \leq & 1 + \int_{1}^{+ \infty} M {exp (p K \int_{0}^{+ \infty} L_{2 s} d s) \geq x} d x \\ = & 1 + \int_{0}^{+ \infty} L_{2 s} d s \int_{0}^{+ \infty} exp (py \int_{0}^{+ \infty} L_{2 s} d s) \\ M {K \geq y} d y \\ \leq & 1 + \int_{0}^{+ \infty} L_{2 s} d s \int_{0}^{+ \infty} exp (py \int_{0}^{+ \infty} L_{2 s} d s) \\ {1 - (2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1)} d y \\ = & 1 + 2 (\int_{0}^{+ \infty} L_{2 s} d s) \int_{0}^{+ \infty} exp (py \int_{0}^{+ \infty} L_{2 s} d s) \\ (1 + exp {(\frac{- π x}{\sqrt{3}})}^{- 1}) d y \\ = & 1 + 2 \int_{1}^{+ \infty} {1 + x^{π / (\sqrt{3} p \int_{0}^{+ \infty} L_{2 s} d s)}}^{- 1} d x \\ < & + \infty . \end{matrix}$ So we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ \leq & lim_{| X_{0} - Y_{0} | \to 0} | X_{0} - Y_{0} |^{p} exp (p \int_{0}^{+ \infty} L_{1 s} d s) \\ E [p exp (K \int_{0}^{+ \infty} L_{2 s} d s)] \\ = & 0 . \end{matrix}$ Hence, it follows from the definition of stability in p-th moment that the multi-dimensional uncertain differential equation is stable in p-th moment under the strong Lipschitz condition. The theorem is proved.

Remark 1. Theorem 4 gives the sufficient condition but not the necessary condition for multi-dimensional uncertain differential equation being stable in p-th moment.

Example 5. Consider a 2-dimensional uncertain differential equation $d X_{t} = (\begin{matrix} - 1 0 \\ 0 - 1 \end{matrix}) d t + (\begin{matrix} 0 - 1 \\ 1 0 \end{matrix}) X_{t} d C_{t} .$ (15) where C_t is a Liu process. Its two solutions with initial values X_t and Y_t are: $X_{t} = (\begin{matrix} exp (- t) sin (C_{t}) exp (- t) cos (C_{t}) \\ - exp (- t) cos (C_{t}) exp (- t) cos (C_{t}) \end{matrix}) X_{0}$ (16) and $Y_{t} = (\begin{matrix} exp (- t) sin (C_{t}) exp (- t) cos (C_{t}) \\ - exp (- t) cos (C_{t}) exp (- t) cos (C_{t}) \end{matrix}) Y_{0},$ (17) respectively. Then we have $\begin{matrix} | X_{t} - Y_{t} | \\ \leq & | (\begin{matrix} exp (- t) sin (C_{t}) - exp (- t) cos (C_{t}) \\ exp (- t) cos (C_{t}) exp (- t) cos (C_{t}) \end{matrix}) | \\ \cdot | X_{0} - Y_{0} | \\ \leq & 2 exp (- t) \cdot | X_{0} - Y_{0} | . \end{matrix}$ So we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ \leq & lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} 2 exp (- pt) | X_{0} - Y_{0} |^{p}] \\ = & lim_{| X_{0} - Y_{0} | \to 0} E [2 | X_{0} - Y_{0} |^{p}] = 0 . \end{matrix}$ almost surely. According to Deffinition 15, we know the 2-dimensional uncertain differential equation is stable in p-th moment. However, the coefficients let $f (t, x) = (\begin{matrix} - 1 0 \\ 0 - 1 \end{matrix}) x$ and $g (t, x) = (\begin{matrix} 0 - 1 \\ 1 0 \end{matrix}) x$ satisfy $| f (t, x) - f (t, y) | \leq | x - y |, \forall x, y \in R^{2}, t \geq 0$ and $| g (t, x) - g (t, y) | \leq | x - y |, \forall x, y \in R^{2}, t \geq 0 .$ where L_1t = 1 and L_2t = 1 are two functions not satisfying the condition in Theorem 4.

Theorem 5. Suppose a linear multi–dimensional uncertain differential equation $d X_{t} = (U_{1 t} X_{t} + U_{2 t}) d t + (V_{1 t} X_{t} + V_{2 t}) d C_{t}$ (18) is stable in p-th moment if and only if the infinite norm of the coefficient matric functions U_it and V_it (i=1,2) are bounded, and $\int_{0}^{+ \infty} | U_{1 t} | d t < + \infty, \int_{0}^{+ \infty} | V_{1 t} | d t < \frac{π}{\sqrt{3} p} .$ (19)

Proof. Assume that X_t and Y_t are two solutions of the linear multi-dimensional uncertain differential equation (18) with two different initial values X₀ and Y₀, respectively, i.e., $\begin{matrix} X_{t} = (U_{1 t} X_{t} + U_{2 t}) d s + (V_{1 t} X_{t} + V_{2 t}) d C_{s}, \\ Y_{t} = (U_{1 t} Y_{t} + U_{2 t}) d s + (V_{1 t} Y_{t} + V_{2 t}) d C_{s} . \end{matrix}$ Then we have $\begin{matrix} d (X_{t} - Y_{t}) \\ = & U_{1 t} (X_{t} - Y_{t}) d s + V_{1 t} (X_{t} - Y_{t}) d C_{s} . \end{matrix}$ Thus, we $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq & | X_{0} - Y_{0} | + | \int_{0}^{t} U_{1 s} (X_{t} (γ) - Y_{t} (γ)) d s | \\ + | \int_{0}^{t} V_{1 s} (X_{t} (γ) - Y_{t} (γ)) d C_{s} (γ) | \\ \leq & | X_{0} - Y_{0} | + \int_{0}^{t} | U_{1 s} (X_{t} (γ) - Y_{t} (γ)) | d s \\ + \int_{0}^{t} | V_{1 s} (X_{t} (γ) - Y_{t} (γ)) | K (γ) d s \\ \leq & | X_{0} - Y_{0} | \\ + \int_{0}^{t} (| U_{1 s} | + K (γ) | V_{1 s} |) | X_{s} (γ) - Y_{s} (γ) | d s \end{matrix}$ where K (γ) is the Lipschitz constants of C_t (γ). By the Gronwall’s inequality, for any t > 0, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq & | X_{0} - Y_{0} | exp (\int_{0}^{t} (| U_{1 s} | + K (γ) | V_{1 s} |) d s) . \end{matrix}$ So we get $\begin{matrix} | X_{t} - Y_{t} | \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} | U_{1 t} | d s) \\ exp (K \int_{0}^{+ \infty} | V_{1 t} | d s) \end{matrix}$ almost surely and we have

$\begin{matrix} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ \leq & | X_{0} - Y_{0} |^{p} exp (p \int_{0}^{+ \infty} | U_{1 t} | d s) \\ E [p exp (K \int_{0}^{+ \infty} | V_{1 t} | d s)] . \end{matrix}$

So we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] \\ \leq & lim_{| X_{0} - Y_{0} | \to 0} | X_{0} - Y_{0} |^{p} exp (p \int_{0}^{+ \infty} | U_{1 t} | d s) \\ E [p exp (K \int_{0}^{+ \infty} | V_{1 t} | d s)] \\ = & 0 \end{matrix}$ if and only if |U_1t| is integrable on [0, + ∞) and and $\int_{0}^{+ \infty} | V_{1 t} | < \frac{π}{\sqrt{3} p}$ . Hence, it follows from the definition of stability in p-th moment that the linear multi-dimensional uncertain differential equation is stable in p-th moment. The theorem is proved.

Theorem 6. If a multi-dimensional uncertain differential equation is stable in p-th moment, then it is stable in measure.

Proof. It follows from the definition of stability in mean of a multi-dimensional uncertain differential equation that for two solutions X_t and Y_t with different initial values X₀ and Y₀, respectively, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p}] = 0 .$ By Markov inequality, for any given real number ε > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M [sup_{t \geq 0} | X_{t} - Y_{t} | \geq ε] \\ \leq & lim_{| X_{0} - X_{0} | \to 0} \frac{E [{sup}_{t \geq 0} | X_{t} - Y_{t} |^{p}]}{ε^{p}} = 0 . \end{matrix}$ So we can obtain $lim_{| X_{0} - Y_{0} | \to 0} M {sup_{t \geq 0} | X_{t} - Y_{t} | \leq ε} = 1$ almost surely. Therefore by the definition of stability in measure that the multi-dimensional uncertain differential equation is stable in measure.

Remark 2. Theorem 4 gives a conclusion for multi-dimensional uncertain differential equation that stability in p-th moment implies stability in measure. Generally, stability in measure does not imply stability in p-th moment.

Theorem 7. For any two real numbers 0< p₁ < p₂ < + ∞. If a multi-dimensional uncertain differential equation is stable in p₂-th moment, then it is stable in p₁-th moment.

Proof. It follows from the definition of stability in p₂-th moment of a multi-dimensional uncertain differential equation that for two solutions X_t and Y_t with different initial values X₀ and Y₀, respectively, we have $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p_{2}}] = 0 .$ By H $\ddot{o}$ lder’s inequality, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p_{1}}] \\ \leq & lim_{| X_{0} - X_{0} | \to 0} {E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p_{1} \cdot \frac{p_{2}}{p_{1}}}]}^{\frac{p_{1}}{p_{2}}} \\ = & lim_{| X_{0} - X_{0} | \to 0} {E [sup_{t \geq 0} | X_{t} - Y_{t} |^{\frac{p_{2}}{p_{2} - p_{1}}}]}^{\frac{p_{2} - p_{1}}{p_{2}}} \\ = & lim_{| X_{0} - X_{0} | \to 0} {E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p_{1}}]}^{\frac{p_{1}}{p_{2}}} . \end{matrix}$ So we can obtain $lim_{| X_{0} - Y_{0} | \to 0} E [sup_{t \geq 0} | X_{t} - Y_{t} |^{p_{1}}] = 0$ almost surely. Therefore by the definition of stability in p-th moment that the multi-dimensional uncertain differential equation is stable in p₁-th moment.

Remark 3. Theorem 7 gives a conclusion for multi-dimensional uncertain differential equation that stability in p₂-th moment implies stability in p₁-th moment when p₁ < p₂.

5 Conclusions

Stability of a multi-dimensional uncertain differential equation plays a very important role. This paper proposed a concept of stability in p-th moment for a multi-dimensional uncertain differential equation. A stability theorem of the multi-dimensional uncertain differential equation being stable in p-th moment was provided. A sufficient and necessary condition for a linear multi-dimensional uncertain differential equation being stable in p-th moment was proved. In addition, this paper discussed the relationships among stability in p-th moment, stability in measure and stability in mean. Some examples were given to illustrate the theoretical considerations. In future, we will extend the work on stability of multi-dimensional uncertain differential equation to high order uncertain differential equation and other types uncertain differential equation.

Compliance with ethical standards

Conflict of Interest: Gang Shi, Xiaohua Li and Lifen Jia declare that they have no conflict of interest.

Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

Footnotes

Acknowledgments

This work was supported by Doctoral Fund of Xinjiang University (No. BS180247).

References

Chen

and Liu

, Existence and uniqueness theorem for uncertain differential equations, Fuzzy Optimization and Decision Making 9 1 (2010), 69–81.

Chen

and Ralescu

D.A.

, Liu process and uncertain calculus, Journal of Uncertainty Analysis and Applications 1, Article 2, (2013).

Gao

, Existence and uniqueness theorem on uncertain differential equations with local lipschitz condition, Journal of Uncertain Systems 6 3 (2012), 223–232.

Ito

, Stochastic integral, Proceedings of the Imperial Academy 20 8 (1944), 519–524.

Ito

, On stochastic differential equations, Memoirs of The American Mathematical Society 4 (1951).

and Zhou

, Multi-dimensional uncertain differential equation existence and uniqueness of the solution, Fuzzy Optimization and Decision Making 14 4 (2015), 477–491.

Kahneman

and Tversky

, Prospect theory: An analysis of decision under risk, Econometrica 47 2 (1979), 263–291.

, Peng

and Zhang

, Multifactor uncertain differential equation, Journal of Uncertainty Analysis and Applications 3 Article 7, (2015).

Liu

, Uncertainty Theory, 2nd Edition, Springer-Verlag, Berlin, 2007.

10.

Liu

, Fuzzy process, hybrid process and uncertain process, Journal of Uncertain Systems 2 1 (2008), 3–16.

11.

Liu

, Some research problems in uncertainty theory, Journal of Uncertain Systems 3 1 (2009), 3–10.

12.

Liu

, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, 2010.

13.

Liu

, Why is there a need for uncertainty theory? Journal of Uncertain Systems 6 1 (2012), 3–10.

14.

Liu

and Yao

, Uncertain integral with respect to multiple canonical processes, Journal of Uncertain Systems 6 4 (2012), 250–255.

15.

Liu

, Uncertainty distribution and independence of uncertain processes, Fuzzy Optimization and Decision Making 13 3 (2014), 259–271.

16.

Liu

, Ke

and Fei

, Almost sure stability for uncertain differential equation, Fuzzy Optimization and Decision Making 13 4 (2014), 463–473.

17.

Liu

, An analytic method for solving uncertain differential equations, Journal of Uncertain Systems 6 4 (2012), 244–249.

18.

Sheng

and Gao

, Exponential stability of uncertain differential equation, Soft Computing 20 9 (2016), 3673–3678.

19.

Sheng

and Wang

, Stability in p-th moment for uncertain differential equation, Journal of Intelligent & Fuzzy Systems 26 3 (2014), 1263–1271.

20.

Sheng

, Shi

and Cui

, Almost sure stability for multifactor uncertain differential equation, Journal of Intelligent & Fuzzy Systems 32 3 (2017), 2187–2194.

21.

Sheng

and Shi

, Stability in mean of multi-dimensional uncertain differential equations, Applied Mathematics and Computation 353 (2019), 178–188.

22.

, Wu

and Zhou

, Stability of multi-dimensional uncertain differential equation, Soft Computing 20 12 (2016), 4991–4998.

23.

Wiener

, Differential-space, Journal of Mathematics and Physics 2 1–4 (1923), 131–174.

24.

Yao

, Extreme values and integral of solution of uncertain differential equation, Journal of Uncertainty Analysis and Applications 1, Article 3(2013).

25.

Yao

, A type of uncertain differential equations with analytic solution, Journal of Uncertainty Analysis and Applications 1 Article 8(2013).

26.

Yao

and Chen

, A numerical method for solving uncertain differential equations, Journal of Intelligent & Fuzzy Systems 25 3 (2013), 825–832.

27.

Yao

, Gao

and Gao

, Some stability theorems of uncertain differential equation, Fuzzy Optimization and Decision Making 12 1 (2013), 3–13.

28.

Yao

, Ke

and Sheng

, Stability in mean for uncertain differential equation, Fuzzy Optimization and Decision Making 14 3 (2015), 365–379.

29.

Yao

, Multi-dimensional uncertain calculus with Liu process, Journal of Uncertain Systems 8 4 (2014), 244–254.

30.

Yang

, Ni

and Zhang

, Stability in inverse distribution for uncertain differential equtions, Journal of Intelligent & Fuzzy Systems 32 (2017), 2051–2059.

31.

Zhang

and Chen

, Multi-dimensional canonical process, Information: An International Interdisciplinary Journal 16 2(A) (2013), 1025–1030.