Almost Sure Stability for Multi-Dimensional Uncertain Differential Equations

Abstract

Multi-dimensional uncertain differential equation is a tool to model an uncertain multi-dimensional dynamic system. Furthermore, stability has a significant role in the field of differential equations because it can be describe the effect of the initial value on the solution of the differential equation. Hence, the concept of almost sure stability is presented concerning multi-dimensional uncertain differential equation in this paper. Moreover, a stability theorem, that is a condition, is derived to judge whether a multi-dimensional uncertain differential equation is almost surely stable or not. Additionally, the paper takes a counterexample to show that the given condition is not necessary for a multi-dimensional uncertain differential equation being almost surely stable.

1. Introduction

Probability theory was established to model randomness associated with frequency. Except for randomness, human uncertainty in relation to belief degree is another type of indeterminacy. For modeling human uncertainty, Liu [1] introduced uncertainty theory on the basis of normality, duality and subadditivity axioms in 2007. Furthermore, Liu [2] refined uncertainty theory by presenting product axiom which is completely different from the one in probability theory. Then some basic concepts were put forward such as uncertain measure for describing belief degrees, uncertain variable for modeling uncertain quantities, uncertainty distribution for describing uncertain variables, and expected value for ranking uncertain variables. Following that, uncertainty theory becomes an almost completely theoretical system, and has been used to many fields such as uncertain reliability analysis [3], vehicle routing problem [4], portfolio optimization [5], and uncertain supply chain management [6].

Uncertain process was first presented by Liu [7] as a tool to model the evolution of uncertain phenomena, which is actually a sequence of uncertain variables varied by time. To describe an uncertain process, Liu [8] put forward a concept of uncertainty distribution. Meanwhile, a sufficient and necessary condition for a function being the uncertainty distribution of an uncertain process was proved in [8]. Besides, a concept of independence of uncertain processes was defined by Liu [8]. Following that, Liu provided the operational law for uncertain processes in [8] and defined stationary independent increment process in [7]. Additionally, Liu [2] proposed Liu process that is an uncertain process with stationary and independent normal uncertain increments. Then, Zhang and Chen [9] generalized it to the multi-dimensional case.

Based on Liu process, uncertain calculus was introduced in [2], and uncertain differential equation was first established by Liu [7] which is a type of differential equation. Then Chen and Liu [10] proved the existence and uniqueness theorem for the solution of an uncertain differential equation under linear growth and Lipschitz conditions. Then Gao [11] gave an existence and uniqueness theorem under linear growth and local Lipschitz conditions. After that, the stability of the solution was studied by many scholars. For example, Liu [2] first investigated stability for the solution of an uncertain differential equation in the sense of uncertain measure. Yao et al. [12] put forward a concept of stability in mean, and gave a sufficient condition. In additon, Liu et al. [13] proposed almost sure stability and gave some theorems. Additionally, some researchers studied numerical solution for an uncertain differential equation, such as Gao [14]. Then, uncertain differential equation has been applied to uncertain fiance [15,16], uncertain optimal control [17].

Consider dynamic system being effected by various kinds of uncertain factors, Yao [18] proposed multi-dimensional uncertain differential equation and proved existence and uniqueness of its solution. The solution depends on initial value and we hope to get stable solution, while stability can be describe the effect of the initial value on the solution of the differential equation. Hence, the property of stability is worth studying. Thus, Su et al. [19] studied the stability of the solution for such differential equations in the sense of uncertain measure. Sheng and Shi [20] studied stability in mean for multi-dimensional uncertain differential equation. However, almost sure stability is not been studied. So, this paper aims at proposing almost sure stability and providing some theorems for the solution of a multi-dimensional uncertain differential equation being almost surely stable. The reminder of the paper is organized as follows. In Section 2, we will review some basic concepts and properties of uncertain variable, uncertain process and uncertain differential equation. Then we will devote Section 3 to proposing a concept of almost sure stability. In Section 4, we will prove a stability theorem. Finally, a brief summary will be made 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, which is necessary for understanding the whole paper.

Let $\Gamma$ be a nonempty set, and $\mathcal{L}$ a $\sigma$-algebra over $\Gamma$. Each element $\Lambda$ in $ℒ$ is called an event and let $ℳ\{\Lambda \}$ to denote the belief degree that $\Lambda$ may happen.

Definition 1

(Liu [1]). An uncertain measure is a function $ℳ$ from $ℒ$ to ℜ under condition that $ℳ$ satisfies the following axioms,

Axiom 1.

$ℳ\{\Gamma \}=1$ for the universal set Γ;

Axiom 2.

$ℳ\{\Lambda \}+ℳ\left\{{\Lambda }^c\right\}=1$ for any event Λ;

Axiom 3.

For ${\Lambda }_i\in \mathcal{L},i=1,2,\cdots$, we have

$$ℳ\left\{\bigcup _{i=1}^{\infty }{\Lambda }_i\right\}\le \sum _{i=1}^{\infty }ℳ\left\{{\Lambda }_i\right\}.$$

The triplet $(\Gamma ,\mathcal{L},ℳ)$ is called an uncertainty space.

Definition 2

(Liu [1]). An uncertain variable is a measurable function ξ from an uncertainty space $(\Gamma ,\mathcal{L},ℳ)$ to the set of real numbers.

Definition 3

(Liu [1]). Suppose ξ is an uncertain variable. Then the uncertainty distribution of ξ is defined by

$$\Phi \left(x\right)=ℳ\left\{\xi \le x\right\}$$

for any real number $x.$

Definition 4

(Liu [7]). Assume that $(\Gamma ,\mathcal{L},ℳ)$ is an uncertainty space and T is an totally-ordered indexed set. If real-valued function $Z_t\left(\gamma \right)$ is measurable where $(\gamma ,t)\in (\Gamma ,\mathcal{L},ℳ)\times T$, then $Z_t$ is an uncertain process.

Definition 5

(Liu [2]). Let $C_t$ be an uncertain process and let it satisfy

(i)

$C_0=0$ and almost all of its sample paths are Lipschitz continuous,

(ii)

increments of $C_t$ are independent and stationary,

(iii)

each increment $C_{s+t}-C_s$ is a normal uncertain variable $\mathcal{N}(0,t)$, that is, the uncertainty distribution of $C_{s+t}-C_s$ is

$$\Phi \left(x\right)={\left(1+exp\left(\frac{-\pi x}{\sqrt{3}t}\right)\right)}^{-1},x\in \Re .$$

Then $C_t$ is said to be a Liu process.

Definition 6

(Liu [2]). Assume that $Z_t$ is an uncertain process and $C_t$ is a Liu process. For any partition of interval $[a,b]$ with $a=t_1<t_2<\cdots <t_{n+1}=b$, the mesh is denoted by

$$\Delta =\underset{1\le i\le n}{max}|t_{i+1}-t_i|.$$

Then Liu integral of $Z_t$ concerning $C_t$ is

$${\int }_a^b{Z}_t\mathrm{d}{C}_t=\underset{\Delta \to 0}{lim}\sum _{i=1}^n{Z}_{t_i}·(C_{t_{i+1}}-C_{t_i})$$

as long as the limit almost surely exists and is finite. In this circumstance, $Z_t$ is called integrable.

Definition 7

(Liu [21]). Assume that $Z_t$ is an uncertain process and $C_t$ is a Liu process. There are two parameters ${\mu }_t$ and ${\nu }_t$ such that

$$Z_t=Z_0+{\int }_0^t{\mu }_s\mathrm{d}s+{\int }_0^t{\nu }_s\mathrm{d}{C}_s,\forall t\ge 0.$$

Then $Z_t$ is said to be a general Liu process with drift ${\mu }_t$ and diffusion ${\nu }_t$. Furthermore, its corresponding uncertain differential is

$$\mathrm{d}{Z}_t={\mu }_t\mathrm{d}t+{\nu }_t\mathrm{d}{C}_t.$$

Lemma 1

(Yao et al. [22]). Assume that $C_t$ is a Liu process. Then there is an uncertain variable $K\left(\gamma \right)$ where $K\left(\gamma \right)$ is a Lipschitz constant of $C_t\left(\gamma \right)$ for every γ, and

$$\underset{x\to +\infty }{lim}ℳ\left\{\gamma \right|K\left(\gamma \right)\le x\}=1.$$

Definition 8

(Zhang and Chen [9]). Let $C_{jt},j=1,2,\cdots ,n$ be independent Liu processes on $(\Gamma ,\mathcal{L},ℳ)$. Then ${\mathit{C}}_t=(C_{1t},C_{2t},\cdots ,C_{nt})$ is said to be an n-dimensional Liu process.

Definition 9

(Yao [18]). Suppose that ${\mathit{C}}_t=(C_{1t},C_{2t},\cdots ,C_{nt})$ is an n-dimensional Liu process, and suppose that ${\mathit{Z}}_t={\left[Z_{ijt}\right]}_{ℳ\times n}$ is an uncertain matrix process with $Z_{ijt}$ being integrable uncertain processes. Then the uncertain integral of $Z_t$ concerning ${\mathit{C}}_t$ is defined by

$${\int }_a^b{\mathit{Z}}_t\mathrm{d}{\mathit{C}}_t=\left[\begin{array}{cc}{\displaystyle \sum _{j=1}^n{\int }_a^b{Z}_{1jt}\mathrm{d}{C}_{jt}}& \\ {\displaystyle \sum _{j=1}^n{\int }_a^b{Z}_{2jt}\mathrm{d}{C}_{jt}}& \\ ⋮\\ {\displaystyle \sum _{j=1}^n{\int }_a^b{Z}_{ℳjt}\mathrm{d}{C}_{jt}}& \end{array}\right].$$

Definition 10

(Yao [18]). Assume that ${\mathit{C}}_t$ is an n-dimensional Liu process. If $\mathit{f}(t,\mathit{z}):T\times {\Re }^m\to {\Re }^m$ is a vector-valued function and $\mathit{h}(t,\mathit{z}):T\times {\Re }^m\to m\times n$ matrix is a matrix-valued function. Then an m-dimensional uncertain differential equation concerning ${\mathit{C}}_t$ is defined by

$$\mathrm{d}{\mathit{Z}}_t=\mathit{f}(t,{\mathit{Z}}_t)\mathrm{d}t+h(t,\mathit{z})\mathrm{d}{\mathit{C}}_t.$$

(1)

Obviously, the solution of Equation (1) is an m-dimensional uncertain process.

3. Almost Sure Stability

In this section, a concept of almost sure stability for a multi-dimensional uncertain differential equation will be proposed. First we explain the meaning of some symbols. For an m-dimensional vector $\mathit{z}=(z_1,z_2,\cdots ,z_m)$ and an $m\times n$ matrix $A=\left[a_{ij}\right]$, we use the infinite norm

$$\left|\mathit{z}\right|=\underset{1\le i\le m}{max}|z_i|,|B|=\underset{1\le i\le m}{max}\sum _{j=1}^n\left|a_{ij}\right|.$$

Definition 11.

Suppose that ${\mathit{Z}}_t$ and ${\mathit{Y}}_t$ are two solutions of the multi-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=\mathit{f}(t,{\mathit{Z}}_t)\mathrm{d}t+\mathit{h}(t,{\mathit{Z}}_t)\mathrm{d}{\mathit{C}}_t$$

(2)

with different initial values ${\mathit{Z}}_0$ and ${\mathit{Y}}_0$. If

$$ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}|{\mathit{Z}}_t-{\mathit{Y}}_t|=0\right\}=1$$

holds for any $t\ge 0$, then Equation (2) is said to be almost surely stable.

Example 1.

Consider the following 2-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=\left[\begin{array}{cc}{\mu }_{1t}& \\ {\mu }_{2t}\end{array}\right]\mathrm{d}t+\left[\begin{array}{ccc}{\nu }_{1t}& 0& \\ 0& {\nu }_{2t}\end{array}\right]\left[\begin{array}{cc}\mathrm{d}{C}_{1t}& \\ \mathrm{d}{C}_{2t}\end{array}\right].$$

(3)

It is obvious that the two solutions of Equation (3) with initial values ${\mathit{Z}}_0$ and ${\mathit{Y}}_0$ are

$${\mathit{Z}}_t={\mathit{Z}}_0+\left[\begin{array}{cc}{\displaystyle {\int }_0^t{\mu }_{1s}\mathrm{d}s+{\int }_0^t{\nu }_{1s}\mathrm{d}{C}_{1t}}& \\ {\displaystyle {\int }_0^t{\mu }_{2s}\mathrm{d}s+{\int }_0^t{\nu }_{2s}\mathrm{d}{C}_{2t}}\end{array}\right]$$

and

$${\mathit{Y}}_t={\mathit{Y}}_0+\left[\begin{array}{cc}{\displaystyle {\int }_0^t{\mu }_{1s}\mathrm{d}s+{\int }_0^t{\nu }_{1s}\mathrm{d}{C}_{1t}}& \\ {\displaystyle {\int }_0^t{\mu }_{2s}\mathrm{d}s+{\int }_0^t{\nu }_{2s}\mathrm{d}{C}_{2t}}\end{array}\right],$$

respectively. Then we have

$$|{\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)|=|{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|$$

for any $t\ge 0,\gamma \in \Gamma$. Since

$$ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}|{\mathit{Z}}_t-{\mathit{Y}}_t|=0\right\}=ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}|{\mathit{Z}}_0-{\mathit{Y}}_0|=0\right\}=1,$$

Equation (3) is almost surely stable according to Definition 11.

Example 2.

Consider the following 2-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=-{\mathit{Z}}_t\mathrm{d}t+\left[\begin{array}{ccc}{\nu }_1& 0& \\ 0& {\nu }_2\end{array}\right]\left[\begin{array}{cc}\mathrm{d}{C}_{1t}& \\ \mathrm{d}{C}_{2t}\end{array}\right].$$

(4)

It is obvious that the two solutions of Equation (4) with initial values ${\mathit{Z}}_0$ and ${\mathit{Y}}_0$ are

$${\mathit{Z}}_t=exp(-t){\mathit{Z}}_0+\left[\begin{array}{cc}{\displaystyle {\int }_0^t{\nu }_{1}exp(s-t)\mathrm{d}{C}_{1s}}& \\ {\displaystyle {\int }_0^t{\nu }_{2}exp(s-t)\mathrm{d}{C}_{2s}}\end{array}\right]$$

and

$${\mathit{Y}}_t=exp(-t){\mathit{Y}}_0+\left[\begin{array}{cc}{\displaystyle {\int }_0^t{\nu }_{1}exp(s-t)\mathrm{d}{C}_{1s}}& \\ {\displaystyle {\int }_0^t{\nu }_{2}exp(s-t)\mathrm{d}{C}_{2s}}\end{array}\right],$$

respectively. Then we have

$$|{\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)|=exp(-t)|{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|\le |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|$$

for any $t\ge 0,\gamma \in \Gamma$. Since

$$ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}|{\mathit{Z}}_t-{\mathit{Y}}_t|=0\right\}=1,$$

Equation (4) is almost surely stable according to Definition 11.

Example 3.

Consider the following 2-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=-{\mathit{X}}_t\mathrm{d}t+\left[\begin{array}{ccc}{\nu }_1& 0& \\ 0& {\nu }_2\end{array}\right]\left[\begin{array}{cc}\mathrm{d}{C}_{1t}& \\ \mathrm{d}{C}_{2t}\end{array}\right].$$

(5)

It is obviously that the two solutions of Equation (5) with initial values ${\mathit{Z}}_0$ and ${\mathit{Y}}_0$ are

$${\mathit{Z}}_t=exp\left(t\right){\mathit{X}}_0+\left[\begin{array}{cc}{\displaystyle {\int }_0^t{\nu }_{1}exp(s-t)\mathrm{d}{C}_{1s}}& \\ {\displaystyle {\int }_0^t{\nu }_{2}exp(s-t)\mathrm{d}{C}_{2s}}\end{array}\right]$$

and

$${\mathit{Y}}_t=exp\left(t\right){\mathit{Y}}_0+\left[\begin{array}{cc}{\displaystyle {\int }_0^t{\nu }_{1}exp(s-t)\mathrm{d}{C}_{1s}}& \\ {\displaystyle {\int }_0^t{\nu }_{2}exp(s-t)\mathrm{d}{C}_{2s}}\end{array}\right],$$

respectively. Then we have

$$|{\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)|=exp\left(t\right)|{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|\to +\infty$$

for any $\gamma \in \Gamma$ as $t\to +\infty$. So Equation (5) is not almost surely stable according to Definition 11.

4. Stability Theorem

In this section, we provide a sufficient condition for a multi-dimensional differential equation being almost surely stable.

Theorem 1.

The multi-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=\mathit{f}(t,Z_t)\mathrm{d}t+\mathit{h}(t,{\mathit{Z}}_t)\mathrm{d}{\mathit{C}}_t$$

(6)

is almost surely stable if the coefficients $\mathit{f}(t,\mathit{z})$ and $\mathit{h}(t,\mathit{z})$ satisfy the strong Lipschitz condition

$$|\mathit{f}(t,\mathit{z})-\mathit{f}(t,\mathit{y})|+|\mathit{h}(t,\mathit{z})-\mathit{h}(t,\mathit{y})|\le L_t|\mathit{z}-\mathit{y}|$$

(7)

for any $\mathit{z},\mathit{y}\in {\Re }^m,t\ge 0,$ where $L_t$ is some positive function satisfying

$${\int }_0^{+\infty }{L}_t\mathrm{d}t<+\infty .$$

(8)

Proof. 

Suppose that ${\mathit{Z}}_t$ and ${\mathit{Y}}_t$ are two solutions of Equation (6) with initial values ${\mathit{Z}}_0$ and ${\mathit{Y}}_0$, respectively. Then for Lipschitz continuous sample paths ${\mathit{C}}_t\left(\gamma \right)$, we have

$${\mathit{Z}}_t\left(\gamma \right)={\mathit{Z}}_0\left(\gamma \right)+{\int }_0^t\mathit{f}(s,{\mathit{Z}}_s\left(\gamma \right))\mathrm{d}s+{\int }_0^t\mathit{h}(s,{\mathit{Z}}_s\left(\gamma \right))\mathrm{d}{\mathit{C}}_s\left(\gamma \right)$$

(9)

and

$${\mathit{Y}}_t\left(\gamma \right)={\mathit{Y}}_0\left(\gamma \right)+{\int }_0^t\mathit{f}(s,{\mathit{Y}}_s\left(\gamma \right))\mathrm{d}s+{\int }_0^t\mathit{h}(s,{\mathit{Y}}_s\left(\gamma \right))\mathrm{d}{\mathit{C}}_s\left(\gamma \right).$$

(10)

It follows from Equations (9) and (10) that

$$\begin{array}{ccc}\hfill {\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)& =& \left[{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)\right]+{\int }_0^t\left[\mathit{f}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{f}(s,{\mathit{Y}}_s\left(\gamma \right))\right]\mathrm{d}s\hfill \\ & & +{\int }_0^t\left[\mathit{h}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{h}(s,{\mathit{Y}}_s\left(\gamma \right))\right]\mathrm{d}{\mathit{C}}_s\left(\gamma \right)\hfill \end{array}$$

Assume $K_j\left(\gamma \right)$ are the Lipschitz constants of $C_{jt}\left(\gamma \right),j=1,2,\cdots ,n$, respectively. We take $K\left(\gamma \right)={\bigvee }_{j=1}^n{K}_j\left(\gamma \right)$. Due to Equation (7), we have

$$\begin{array}{ccc}\hfill |{\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)|& =& |\left[{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)\right]+{\int }_0^t\left[\mathit{f}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{f}(s,{\mathit{Y}}_s\left(\gamma \right))\right]\mathrm{d}s\hfill \\ & & +{\int }_0^t\left[\mathit{h}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{h}(s,{\mathit{Y}}_s\left(\gamma \right))\right]\mathrm{d}{\mathit{C}}_s\left(\gamma \right)|\hfill \\ & \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|+{\int }_0^t|\mathit{f}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{f}(s,{\mathit{Y}}_s\left(\gamma \right))|\mathrm{d}s\hfill \\ & & +\left|{\int }_0^t\left[\mathit{h}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{h}(s,{\mathit{Y}}_s\left(\gamma \right))\right]\mathrm{d}{\mathit{C}}_s\left(\gamma \right)\right|\hfill \\ & \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|+{\int }_0^t|\mathit{f}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{f}(s,{\mathit{Y}}_s\left(\gamma \right))|\mathrm{d}s\hfill \\ & & +{\int }_0^t|\mathit{h}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{h}(s,{\mathit{Y}}_s\left(\gamma \right))|\left|\mathrm{d}{\mathit{C}}_s\left(\gamma \right)\right|\hfill \\ & \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|+{\int }_0^t{L}_s|{\mathit{Z}}_s\left(\gamma \right)-{\mathit{Y}}_s\left(\gamma \right)|\mathrm{d}s\hfill \\ & & +{\int }_0^{t}K\left(\gamma \right)|\mathit{h}(s,{\mathit{Z}}_s\left(\gamma \right))-\mathit{h}(s,{\mathit{Y}}_s\left(\gamma \right))|\mathrm{d}s\hfill \\ & \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|+{\int }_0^t{L}_s|{\mathit{Z}}_s\left(\gamma \right)-{\mathit{Y}}_s\left(\gamma \right)|\mathrm{d}s\hfill \\ & & +{\int }_0^{t}K\left(\gamma \right)L_s|{\mathit{Z}}_s\left(\gamma \right)-{\mathit{Y}}_s\left(\gamma \right)|\mathrm{d}s\hfill \\ & \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|+(1+K\left(\gamma \right)){\int }_0^t{L}_s|{\mathit{Z}}_s\left(\gamma \right)-{\mathit{Y}}_s\left(\gamma \right)|\mathrm{d}s\hfill \end{array}$$

for any $t\ge 0,\gamma \in \Gamma$. From Gronwall’s inequality, we obtain

$$\begin{array}{ccc}\hfill |{\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)|& \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|exp\left((1+K\left(\gamma \right)){\int }_0^t{L}_s\mathrm{d}s\right)\hfill \\ & \le & |{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|exp\left((1+K\left(\gamma \right)){\int }_0^{+\infty }{L}_s\mathrm{d}s\right)\hfill \end{array}$$

for any $t\ge 0,\gamma \in \Gamma$. Since $K\left(\gamma \right)$ is finite, it follows from Equation (8) that $|{\mathit{Z}}_t\left(\gamma \right)-{\mathit{Y}}_t\left(\gamma \right)|\to 0$ as long as $|{\mathit{Z}}_0\left(\gamma \right)-{\mathit{Y}}_0\left(\gamma \right)|\to 0$, which implies that

$$ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}|{\mathit{Z}}_t-{\mathit{Y}}_t|=0\right\}=1$$

for any $t\ge 0$. According to Definition 11, the multi-dimensional uncertain differential equation is almost surely stable. □

Example 4.

Consider the following 2-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=-exp(-t){\mathit{Z}}_t\mathrm{d}t+\left[\begin{array}{ccc}0& exp(-t)& \\ exp(-t)& 0\end{array}\right]{\mathit{Z}}_t\mathrm{d}{C}_t.$$

(11)

Let $\mathit{f}(t,\mathit{z})=-exp(-t)\mathit{z},\mathit{h}(t,\mathit{z})=\left[\begin{array}{ccc}0& exp(-t)& \\ exp(-t)& 0\end{array}\right]\mathit{z}$ in Theorem 1. Since

$$\left|\mathit{f}\right(t,\mathit{z})-\mathit{f}(t,\mathit{y}\left)\right|+\left|\mathit{h}\right(t,\mathit{z})-\mathit{h}(t,\mathit{y}\left)\right|=2exp(-t)|\mathit{z}-\mathit{y}|$$

and

$${\int }_0^{+\infty }2exp(-t)\mathrm{d}t=2<+\infty .$$

It is obvious that the Equation (11) is almost surely stable according to Theorem 1.

Example 5.

Consider the following 2-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=\left[\begin{array}{ccc}-1& 0& \\ 0& -1\end{array}\right]{\mathit{Z}}_t\mathrm{d}t+\left[\begin{array}{ccc}0& -1& \\ 1& 0\end{array}\right]{\mathit{Z}}_t\mathrm{d}{C}_t.$$

(12)

Let $\mathit{f}(t,\mathit{z})=\left[\begin{array}{ccc}-1& 0& \\ 0& -1\end{array}\right]\mathit{z},\mathit{h}(t,\mathit{z})=\left[\begin{array}{ccc}0& -1& \\ 1& 0\end{array}\right]\mathit{z}$ in Theorem 1. Since

$$\left|\mathit{f}\right(t,\mathit{z})-\mathit{f}(t,\mathit{y}\left)\right|+\left|\mathit{h}\right(t,\mathit{z})-\mathit{h}(t,\mathit{y}\left)\right|=2|\mathit{z}-\mathit{y}|$$

and

$${\int }_0^{+\infty }2\mathrm{d}t=+\infty .$$

It is obvious that Equation (12) is not almost surely stable according to Theorem 1. However, the solutions of (12) with initial values ${\mathit{Z}}_0$ and ${\mathit{Y}}_0$ are

$${\mathit{Z}}_t=\left[\begin{array}{ccc}exp(-t)sin\left(C_t\right)& exp(-t)cos\left(C_t\right)& \\ -exp(-t)cos\left(C_t\right)& exp(-t)sin\left(C_t\right)\end{array}\right]{\mathit{Z}}_0$$

$${\mathit{Y}}_t=\left[\begin{array}{ccc}exp(-t)sin\left(C_t\right)& exp(-t)cos\left(C_t\right)& \\ -exp(-t)cos\left(C_t\right)& exp(-t)sin\left(C_t\right)\end{array}\right]{\mathit{Y}}_0,$$

respectively. Then we have

$$\begin{array}{ccc}\hfill |{\mathit{Z}}_t-{\mathit{Y}}_t|& =& \left|\left[\begin{array}{ccc}exp(-t)sin\left(C_t\right)& exp(-t)cos\left(C_t\right)& \\ -exp(-t)cos\left(C_t\right)& exp(-t)sin\left(C_t\right)\end{array}\right]({\mathit{Z}}_0-{\mathit{Y}}_0)\right|\hfill \\ & =& exp(-t)\left|\left[\begin{array}{cc}sin\left(C_t\right)(Z_{10}-Y_{10})+cos\left(C_t\right)(Z_{20}-Y_{20})& \\ cos\left(C_t\right)(Z_{10}-Y_{10})+sin\left(C_t\right)(Z_{20}-Y_{20})\end{array}\right]\right|\hfill \\ & \le & 2exp(-t)|{\mathit{Z}}_0-{\mathit{Y}}_0|.\hfill \end{array}$$

Hence we have

$$ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}|{\mathit{Z}}_t-{\mathit{Y}}_t|=0\right\}\ge ℳ\left\{\underset{|{\mathit{Z}}_0-{\mathit{Y}}_0|\to 0}{lim}2exp(-t)|{\mathit{Z}}_0-{\mathit{Y}}_0|=0\right\}=1$$

for any $t\ge 0$. According to Definition 11, we know that the solution of Equation (3) is almost surely stable.

Remark 1.

Considering Example 5, we can assert that the conditions given in Theorem 1 are just sufficient but not necessary for multi-dimensional uncertain differential equations.

Corollary 1.

Assume that ${\mathit{U}}_t$, ${\mathit{V}}_t$ are $n\times n$ matrix-valued functions, and ${\mathit{A}}_t,{\mathit{B}}_t$ are n-dimensional vector-valued functions. Then the linear n-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=({\mathit{U}}_t{\mathit{Z}}_t+{\mathit{A}}_t)\mathrm{d}t+({\mathit{V}}_t{\mathit{Z}}_t+{\mathit{B}}_t)\mathrm{d}{C}_t$$

(13)

is almost surely stable if

$${\int }_0^{+\infty }\left(\right|{\mathit{U}}_t|+|{\mathit{V}}_t\left|\right)\mathrm{d}t<+\infty .$$

(14)

Proof. 

Take $\mathit{f}(t,\mathit{z})={\mathit{U}}_t\mathit{z}+{\mathit{A}}_t$ and $\mathit{h}(t,\mathit{z})={\mathit{V}}_t\mathit{z}+{\mathit{B}}_t$ in Theorem 1. Then we have

$$\begin{array}{ccc}\hfill |\mathit{f}(t,\mathit{z})-\mathit{f}(t,\mathit{y})|+|\mathit{h}(t,\mathit{z})-\mathit{h}(t,\mathit{y})|& =& |{\mathit{U}}_t\mathit{z}-{\mathit{U}}_t\mathit{y}|+|{\mathit{V}}_t\mathit{z}-{\mathit{V}}_t\mathit{y}|\hfill \\ & \le & \left(|{\mathit{U}}_t|+|{\mathit{V}}_t|\right)|\mathit{z}-\mathit{y}|.\hfill \end{array}$$

It follows from Equation (14) and Theorem 1 that the linear Equation (13) is almost surely stable. Thus, the proof is finished. □

Example 6.

Consider the following 2-dimensional uncertain differential equation

$$\mathrm{d}{\mathit{Z}}_t=\left[\begin{array}{ccc}\frac{1}{1+t^2}-1& 1& \\ 0& \frac{1}{1+t^2}-1\end{array}\right]{\mathit{Z}}_t\mathrm{d}t+\left[\begin{array}{ccc}\frac{sint}{t}& 0& \\ 0& \frac{sint}{t}\end{array}\right]{\mathit{Z}}_t\mathrm{d}{C}_t.$$

(15)

Let

$$\mathit{f}(t,\mathit{z})=\left[\begin{array}{ccc}\frac{1}{1+t^2}-1& 1& \\ 0& \frac{1}{1+t^2}-1\end{array}\right]\mathit{z},\mathit{h}(t,\mathit{z})=\left[\begin{array}{ccc}\frac{sint}{t}& 0& \\ 0& \frac{sint}{t}\end{array}\right]\mathit{z}$$

in Corollary 1. Since

$${\int }_0^{+\infty }\left(\frac{1}{1+t^2}+\left|\frac{sint}{t}\right|\right)\mathrm{d}t=\frac{\pi }{2}+\frac{\pi }{2}<\infty ,$$

it is obvious that Equation (15) is almost surely stable according to Corollary 1.

5. Conclusions

The multi-dimensional uncertain differential equation an important type of uncertain differential equation driven by multiple Liu process to model uncertain multi-dimensional dynamic phenomena. The solution of the motion of matter described by differential equations is closely dependent on the initial value. In fact, errors and interferences inevitably occur in the calculation and determination of initial values. If the solution of the equation used to model the motion of matter is not stable. That is, the small error or interference of the initial value will lead to the serious consequence. So a stable solution is meaningful to us.

Hence, we mainly studied the stability of multi-dimensional uncertain differential equations and proposed the concept of almost sure stability. Then we proved the stability theorem that is a condition for a multi-dimensional uncertain differential equation being almost surely stable under strong Lipschitz conditions and the condition is not necessary. Especially, we studied the almost sure stability of linear multi-dimensional uncertain differential equations. Additionally, some examples were presented to illustrate how to use the concept and stability theorem to judge whether an multi-dimensional uncertain differential equation being almost surely stable. The results presented in this paper could be used to study the stability of multi-factor uncertain differential equations and high-order uncertain differential equations.

6. Patents

This section is not mandatory, but may be added if there are patents resulting from the work reported in this manuscript.