Stability analysis for uncertain differential equation by Lyapunov’s second method

Abstract

Uncertain differential equation is a type of differential equation driven by Liu process that is the counterpart of Wiener process in the framework of uncertainty theory. The stability theory is of particular interest among the properties of the solutions to uncertain differential equations. In this paper, we introduce the Lyapunov’s second method to study stability in measure and asymptotic stability of uncertain differential equation. Different from the existing results, we present two sufficient conditions in sense of Lyapunov stability, where the strong Lipschitz condition of the drift is no longer indispensable. Finally, illustrative examples are examined to certify the effectiveness of our theoretical findings.

Appendix: Uncertainty theory

In this section, we introduce uncertainty theory including Liu process and uncertain differential equation briefly. The reader may refer to Liu (2011) for details.

Definition 3

(Liu 2009) An uncertain process $C_t$ is said to be a Liu process if

• (i) $C_0=0$ 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 $t^2$, whose uncertainty distribution is

  $$\begin{aligned} \varPhi _t(x)=\left( 1+\exp \left( -\frac{\pi x}{\sqrt{3}t}\right) \right) ^{-1},~~x\in \mathfrak {R}. \end{aligned}$$

Contrast to Wiener process, almost all the sample paths of a Liu process are Lipschitz continuous, and the corresponding Lipschitz constant for all sample paths is essentially a uncertain variable.

Theorem 5

(Yao et al. 2013) Let $C_t$ be a Liu process on an uncertainty space $(\varGamma ,{\mathcal {L}},{\mathcal {M}})$. Then there exits an uncertain variable K such that $K(\gamma )$ is a Lipschitz constant of the sample path $C_t(\gamma )$ for each $\gamma$, and

$$\begin{aligned} {\mathcal {M}}\{\gamma \in \varGamma |K(\gamma )\le x\}\ge 2\left( 1+exp(-\frac{\pi x}{\sqrt{3}})\right) ^{-1}-1. \end{aligned}$$

Then the definition of uncertain differential equation and corresponding stability concepts are provided as follows.

Definition 4

(Liu 2008) Suppose that $C_t$ is a Liu process, and f and g are two measurable real functions. Then

$$\begin{aligned} \mathrm {d}X_t=f(t,X_t)\mathrm {d}t+g(t,X_t)\mathrm {d}C_t, \end{aligned}$$

is called an uncertain differential equation.

Definition 5

(Liu 2009) The uncertain differential Eq. (1) is said to be stable in measure if for any two solutions $X_t$ and $Y_t$ with different initial values $X_0$ and $Y_0$, and for any given number $\varepsilon >0$, we have

$$\begin{aligned} \lim \limits _{|X_0-Y_0|\rightarrow 0}{\mathcal {M}}\left\{ \sup \limits _{t\ge 0}|X_t-Y_t|>\varepsilon \right\} =0. \end{aligned}$$

Theorem 6

(Yao et al. 2013) The uncertain differential Eq. (1) is said to be stable in measure if the coefficients f(t, x) and g(t, x) satisfy the linear growth condition

$$\begin{aligned} |f(t,x)|+|g(t,x)|\le R(1+|x|),~~\forall x\in \mathfrak {R},\,t\ge 0 \end{aligned}$$

for some constant R, and the strong Lipschitz condition

$$\begin{aligned} |f(t,x)-f(t,y)|+|g(t,x)-g(t,y)|\le L_t|x-y|,~~\forall x,y\in \mathfrak {R},\,t\ge 0 \end{aligned}$$

for some integrable function $L_t$ on $[0,+\infty )$.