Stability in p -th moment for uncertain spring vibration equation

Abstract

Uncertain Differential equations are a type of differential equations driven by the Liu processes rather than the Wiener processes. Depending on the order of differentials it contains, an uncertain differential equation could be classified into first-order uncertain differential equation, second-order uncertain differential equation, third-order uncertain differential equation, and so on. The concepts of stability in various senses for the uncertain differential equations could be specified and applied to the uncertain spring vibration differential equations. However, to the best knowledge of mine, many types of stability have been proposed for first-order uncertain differential equations, for example, stability in mean, stability in p-th moment, stability in distribution, almost sure stability and exponential stability. However, stability in measure and stability in mean of high-order uncertain differential equations have been proposed. But only the concept of stability in mean and the concept of stability in measure have been proposed for high-order uncertain differential equations. In this paper, following the concept of stability in p-th moment for first-order uncertain differential equations, we present the concept of stability in p-th moment for general uncertain spring vibration differential equations which are a type of second-order uncertain differential equations.

Keywords

Spring vibration equation stability uncertain differential equation uncertain process uncertainty theory

1 Introduction

Based on normality, duality, subadditivity and product axioms of uncertain measure, uncertainty theory as a mathematical system was founded and developed by Liu [9, 11]. As an important contribution to uncertainty theory, uncertain processes were proposed by Liu in order to model the evolution of uncertain phenomena [10]. An uncertain process is essentially a sequence of uncertain variables indexed by time, that is, it is an uncertain variable at each time. In order to deal with white noise, Liu introduced Liu process as a supplement of Wiener process [11]. A Liu process is a stationary and independent increment process whose increments are normal uncertain variables and almost all sample paths are Lipschitz continuous. Based on such a process, uncertain calculus theory was proposed by Liu to deal with differentiation and integration of uncertain processes [11].

First-order uncertain differential equation is driven by Liu process proposed by Liu [10]. Chen and Liu first proved the existence and uniqueness theorem for an first-order uncertain differential equation [1]. Following that, Liu proposed the concept of stability of first-order uncertain differential equation [11], and Yao et al. verified a series of stability theorems [26]. Moreover, stability in mean, stability in moment, stability in distribution, almost sure stability, and exponential stability for first-order uncertain differential equation have been investigated such as [13–15 , 27]. More importantly, Yao and Chen proved that the solution of a first-order uncertain differential equation can be represented by a family of solutions of ordinary differential equations [25]. High-order uncertain differential equation was first proposed by Yao [24]. Following that, Sheng [16] investigated the stability in measure and stability in mean of high-order uncertain differential equation, Ji and Zhou [8] designed a Runge-Kutta method to solve the high-order uncertain differential equation. By now, uncertain differential equations have been widely applied in many fields such as finance ([12, 18]), optimal control ([28]), differential game ([17]), heat conduction ([19 , 23]), string vibration ([2, 3]), and spring vibration ([4 –7]). In order to further explore the development and applications of uncertain differential equation, the interested readers may refer to the book [24].

Uncertain spring vibration equations were first proposed by Jia and Dai [4] which a special type of second-order uncertain differential equations. The equation describes spring vibration whose external force is affected by an uncertain interference, and the solution and inverse uncertainty distribution of the solution of an uncertain spring vibration equation were derived. Besides, Jia and Yang proved an existence and uniqueness theorem [5] of general uncertain spring vibration equation. Jia and Chen proposed the concepts of stable in measure and stable in mean [7] of general uncertain spring vibration equation. In this paper, following the concept of stability in p-th moment for first-order uncertain differential equations, we present the concept of stability in p-th moment for the uncertain spring vibration differential equations which are a type of second-order uncertain differential equation. We also prove a sufficient condition of stability in p-th moment for general uncertain spring vibration equation. A deep connection among stability in measure and in p-th moment of general uncertain spring vibration equation are derived. The rest of this paper is structured as follows. Section 2 introduces uncertain spring vibration equation. Section 3 proposes stability in p-th moment of general uncertain spring vibration equation, and proves some stability theorems. A comparison of stability of general uncertain spring vibration equation is given in Section 4. Section 5 gives some conclusions.

2 Uncertain spring vibration equation

As a type of second-order differential equation, spring vibration equation describes the vibration of a spring with a time varying external force acting on it. However, in practice, the external force is often affected by interference of noise. In order to model the noise, two processes can be employed, one is Wiener process based on probability theory, the other is Liu process based on uncertainty theory. If we use Wiener process to describe the noise, then the vibration equation becomes a stochastic vibration equation. Nevertheless, Jia and Dai [4] proved that it is not realistic to model the vibration as a stochastic vibration equation in some cases. For example, if we employ stochastic vibration equation $\frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t) + σ (t) \dot{W_{t}}$ to describe vibration of spring, where $\dot{W_{t}} = W_{t} / d t$ and W_t is a Wiener process, it will leads to at least one of velocity and accelerated velocity being equal to ∞ at any time t. Thus, Jia and Dai [4] proposed a general uncertain spring vibration equation whose noise of external force is described by a Liu process. The equation is as follows, $\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t, X_{t}) + σ (t, X_{t}) \dot{C_{t}}, \\ t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] \end{matrix}$ (1) where 2δ is the ratio of c/m, c is damping constant, m is the mass of the vibration object, ω² is the ratio of K/m, K is Hooke’s constant, X_t is displacement, $\dot{C_{t}}$ is the ratio of dC_t/dt denotes the time white noise, C_t is a Liu process, f (t, X_t) is external force, σ (t, X_t) is the diffusion term of external force, and l^e, l^l,l^s are the lengths of equilibrium position, longest, and shortest of the spring, assumes that the initial displacement and velocity are zero, respectively. Jia and Yang [5] proved an existence and uniqueness theorem, Jia et al. [6] designed a numerical method for solving uncertain spring vibration equation, and Jia and Chen [7] proposed the concepts of stability in measure and stability in mean.

Definition 2.1. (Jia and Chen [7]) The general uncertain spring vibration equation (1) is said to be stable in measure if, for any given number ɛ > 0, the solutions X_t and Y_t satisfy $lim_{| X_{0} - Y_{0} | \to 0} M {| X_{t} - Y_{t} | \leq ɛ} = 1, \forall t > 0$ with different initial positions and velocities X₀, Y₀, $\frac{d X_{t}}{d t} |_{t = 0} = 0$ , $\frac{d Y_{t}}{d t} |_{t = 0} = 0, \frac{π}{\sqrt{3} p}$ .

3 Stability in p-th moment

This section we will give a definition of stability in p-th moment for uncertain spring vibration equation. Besides, we give two theorems for general uncertain differential equation being stable in p-th moment.

Definition 3.1. The general uncertain spring vibration equation (1) is said to be stable in p-th moment, 0≤ p ≤ + ∞, if there are two solutions X_t and Y_t satisfy $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p}] = 0, \forall t > 0$ with different initial positions X₀, Y₀ and velocities $\frac{d X_{t}}{d t} |_{t = 0} = 0$ , $\frac{d Y_{t}}{d t} |_{t = 0} = 0$ .

Remark 3.1. In particular, if p = 1, then stable in p-th moment is just stable in mean.

Next we give an example of uncertain spring vibration equation which is stable in p - th moment.

Example 4.1: Consider linear uncertain spring vibration equation $\begin{matrix} \begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 2 δ \frac{d X_{t}}{d t} + ω^{2} X_{t} = f (t) + σ (t) \dot{C_{t}}, \\ t > 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] . \end{matrix} \end{matrix}$ (2) Based on the result of Jia and Yang [5], we have a linear uncertain spring vibration equation (2) has a unique solution for each given initial state. Assume that there are two initial positions and velocities X₀, Y₀, $\frac{d X_{t}}{d t} |_{t = 0} = 0$ , $\frac{d Y_{t}}{d t} |_{t = 0} = 0$ .

The argument breaks down into two cases. Case I: Assume δ > ω. Then for a Lipschitz continuous sample path C_t (γ), we have $\begin{matrix} X_{t} (γ) = (\frac{1}{2} X_{0} + \frac{δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} X_{0} - \frac{δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) f (s) d s \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) σ (s) \dot{C_{s}} (γ) d s, \end{matrix}$ $\begin{matrix} Y_{t} (γ) = (\frac{1}{2} Y_{0} + \frac{δ Y_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} X_{0} - \frac{δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) f (s) d s \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) σ (s) \dot{C_{s}} (γ) d s . \end{matrix}$ Then $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ = | (\frac{1}{2} (X_{0} - Y_{0}) + \frac{δ (X_{0} - Y_{0})}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} (X_{0} - Y_{0}) - \frac{δ (X_{0} - Y_{0})}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} | \\ \leq | (\frac{1}{2} (X_{0} - Y_{0}) + \frac{δ (X_{0} - Y_{0})}{2 \sqrt{δ^{2} - ω^{2}}}) \\ + (\frac{1}{2} (X_{0} - Y_{0}) - \frac{δ (X_{0} - Y_{0})}{2 \sqrt{δ^{2} - ω^{2}}}) | \\ = | X_{0} - Y_{0} | . \end{matrix}$ That is, $| X_{t} - Y_{t} | \leq | X_{0} - Y_{0} | .$ Taking p-th moment on both sides, we have $E [| X_{t} - Y_{t} |^{p}] \leq E [| X_{0} - Y_{0} |^{p}] .$ It is clear that $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p}] = 0 .$ Thus when δ > ω, the uncertain spring vibration equation (2) is stable in p-th moment.

Case II: Assume δ < ω. Then for a Lipschitz continuous sample path C_t (γ), we have

$\begin{matrix} X_{t} (γ) = X_{0} e^{- δ t} cos (\sqrt{ω^{2} - δ^{2}} t) \\ + \frac{δ X_{0}}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (\sqrt{ω^{2} - δ^{2}} t) \end{matrix}$ $\begin{matrix} + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin (\sqrt{ω^{2} - δ^{2}} (t - s)) \\ f (s) d s \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin (\sqrt{ω^{2} - δ^{2}} (t - s)) \\ σ (s) \dot{C_{s}} (γ) d s, \end{matrix}$ and $\begin{matrix} Y_{t} (γ) = Y_{0} e^{- δ t} cos (\sqrt{ω^{2} - δ^{2}} t) \\ + \frac{δ Y_{0}}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (\sqrt{ω^{2} - δ^{2}} t) \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin (\sqrt{ω^{2} - δ^{2}} (t - s)) \\ f (s) d s \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin (\sqrt{ω^{2} - δ^{2}} (t - s)) \\ σ (s) \dot{C_{s}} (γ) d s . \end{matrix}$ Then $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | = | (X_{0} - Y_{0}) e^{- δ t} cos (\sqrt{ω^{2} - δ^{2}} t) \\ + \frac{δ (X_{0} - Y_{0})}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (\sqrt{ω^{2} - δ^{2}} t) | \\ \leq | X_{0} - Y_{0} | (1 + \frac{δ}{\sqrt{ω^{2} - δ^{2}}}) . \end{matrix}$ That is, $\begin{matrix} | X_{t} - Y_{t} | \leq | X_{0} - Y_{0} | (1 + \frac{δ}{\sqrt{ω^{2} - δ^{2}}}) . \end{matrix}$ It is clear that $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p}] = 0 .$ Thus when δ < ω, the uncertain spring vibration equation (2) is stable in p-th moment.

Theorem 3.1. For any two positive real numbers p₁ < p₂, if a general uncertain spring vibration equation (1) is stable in p₂-th moment, then it is stable in p₁-th moment.

Proof: By Definition 3, when a general uncertain spring vibration equation (1) is stable in p₂-th moment, then for two solutions X_t and Y_t with different initial positions X₀, Y₀ and velocities $\frac{d X_{t}}{d t} |_{t = 0} = 0$ , $\frac{d Y_{t}}{d t} |_{t = 0} = 0$ , we have $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p_{2}}] = 0, \forall t > 0 .$ Let ξ = |X_t - Y_t|^{p
₁}, η = 1, $p = \frac{p_{2}}{p_{1}}$ , $q = \frac{p_{2}}{p_{2} - p_{1}}$ . According to Hölder’s inequality, we have $\begin{matrix} E [| X_{t} - Y_{t} |^{p_{1}}] \\ = E [| ξ η |] \leq \sqrt[p]{E | ξ |^{p}} \sqrt[q]{E | η |^{q}} \\ = \sqrt[p]{E | ξ |^{p}} = \sqrt[p_{2}]{(E | X_{t} - Y_{t} |^{p_{1} p})^{p_{1}}} \\ = \sqrt[p_{2}]{(E | X_{t} - Y_{t} |^{p_{2}})^{p_{1}}} . \end{matrix}$ Thus a general uncertain spring vibration equation (1) stable in p₂-th moment implies stable in p₁-th moment. The theorem is completed. □

Theorem 3.2. The general uncertain spring vibration equation (1) is stable in p-th moment if the coefficients f (t, x) and σ (t, x) satisfy the linear growth condition $\begin{matrix} | f (t, x) | + | σ (t, x) | \leq L (1 + | x |), \\ \forall x \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ where L is a constant, and satisfy the strong Lipschitz condition $\begin{matrix} | f (t, x) - f (t, y) | \leq L_{1 t} | x - y |, \\ | σ (t, x) - σ (t, y) | \leq L_{2 t} | x - y |, \\ \forall x \in [l^{s} - l^{e}, l^{l} - l^{e}], t > 0 \end{matrix}$ where L_1t and L_2t are some bounded functions satisfying $\int_{0}^{+ \infty} L_{1 s} d s < + \infty, and \int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3} p} .$ $\int_{0}^{+ \infty} L_{1 s} \int_{d}^{s} \begin{matrix} < + \infty, and \int_{0}^{+ \infty} L_{2 s} \int_{d}^{s} < \frac{π \sqrt{| δ^{2} - ω^{2} |}}{\sqrt{3} p} \end{matrix}$

Proof: According to Jia and Yang [5], the general uncertain spring vibration equation has a unique solution. Suppose that X_t and Y_t are two solutions of the general uncertain spring vibration equation (1) with different initial positions X₀, Y₀, and velocities $\frac{d X_{t}}{d t} |_{t = 0} = 0$ , $\frac{d Y_{t}}{d t} |_{t = 0} = 0$ . The argument breaks down into two cases.

Case I: Assume δ > ω. For a Lipschitz continuous sample path C_t (γ), we have $\begin{matrix} X_{t} (γ) = (\frac{1}{2} X_{0} + \frac{δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} X_{0} - \frac{δ X_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) f (s, X_{s} (γ)) d s \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) σ (s, X_{s} (γ)) d C_{s} (γ), \end{matrix}$ and $\begin{matrix} Y_{t} (γ) = (\frac{1}{2} Y_{0} + \frac{δ Y_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} Y_{0} - \frac{δ Y_{0}}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) f (s, Y_{s} (γ)) d s \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) (t - s)} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) (t - s)}) σ (s, Y_{s} (γ)) d C_{s} (γ), \end{matrix}$ By the strong Lipschitz condition, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | (\frac{1}{2} | X_{0} - Y_{0} | + \frac{δ | X_{0} - Y_{0} |}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t + \sqrt{δ^{2} - ω^{2}} t} \\ + (\frac{1}{2} | X_{0} - Y_{0} | - \frac{δ | X_{0} - Y_{0} |}{2 \sqrt{δ^{2} - ω^{2}}}) e^{- δ t - \sqrt{δ^{2} - ω^{2}} t} | \\ + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) s} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) s}) (f (s, X_{s} (γ)) \\ - f (s, Y_{s} (γ))) | d s \end{matrix}$ $\begin{matrix} + \frac{1}{2 \sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (e^{(- δ + \sqrt{δ^{2} - ω^{2}}) s} \\ - e^{(- δ - \sqrt{δ^{2} - ω^{2}}) s}) (σ (s, X_{s} (γ)) \\ - σ (s, Y_{s} (γ))) | | d C_{s} (γ) | \\ \leq | (\frac{1}{2} | X_{0} - Y_{0} | + \frac{δ | X_{0} - Y_{0} |}{2 \sqrt{δ^{2} - ω^{2}}}) \\ + (\frac{1}{2} | X_{0} - Y_{0} | - \frac{δ | X_{0} - Y_{0} |}{2 \sqrt{δ^{2} - ω^{2}}}) | \\ + \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (f (s, X_{s} (γ)) - f (s, X_{s} (γ))) | d s \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (σ (s, X_{s} (γ)) - σ (s, X_{s} (γ))) | d s \end{matrix}$ $\begin{matrix} \leq | X_{0} - Y_{0} | + \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{1 s} | (X_{s} (γ) - X_{s} (γ) | d s \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{2 s} | X_{s} (γ) - X_{s} (γ) | d s \\ = \frac{(L_{1 s} + K (γ) L_{2 s})}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (X_{s} (γ) - X_{s} (γ) | d s \\ + | X_{0} - Y_{0} | \end{matrix}$ where K (γ) is the Lipschitz constant of C_t (γ). It follows from Gronwall’s inequality that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{t} L_{1 s} d s) exp (\frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{2 s} d s) \\ \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} d s) \\ exp (K (γ) \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ for any t ≥ 0. Thus we have $\begin{matrix} | X_{t} - Y_{t} | \leq | X_{0} - Y_{0} | exp (\int_{0}^{+ \infty} L_{1 s} d s) \\ exp (\frac{K}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ almost surely, where K is a nonnegative uncertain variable such that $M {γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$ Taking p-th moment on both sides, we obtain $\begin{matrix} E [| X_{t} - Y_{t} |^{p}] \leq | X_{0} - Y_{0} |^{p} exp (p \int_{0}^{+ \infty} L_{1 s} d s) \\ E [exp (p \frac{K}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{2 s} d s)], \forall t > 0 . \end{matrix}$ Since $\int_{0}^{+ \infty} L_{1 s} d s < + \infty,$ we have $exp (p \int_{0}^{+ \infty} L_{1 s} d s) < + \infty .$ As for $E [exp (p \frac{K}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{2 s} d s)],$ we set $h = \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3} p}$ for simplicity. By the definition of the expected value, we have $\begin{matrix} E [exp (hpK)] \\ = \int_{0}^{+ \infty} M {exp (hpK) \geq x} d x \\ = \int_{0}^{1} M {exp (hpK) \geq x} d x \\ + \int_{1}^{+ \infty} M {exp (hpK) \geq x} d x \\ \leq 1 + \int_{1}^{+ \infty} M {exp (hpK) \geq x} d x \\ = 1 + h \int_{0}^{+ \infty} exp (hpy) M {K \geq y} d y \\ \leq 1 + h \int_{0}^{+ \infty} exp (hpy) (1 - (2 (1 \\ {+ exp (\frac{- π y}{\sqrt{3}}))}^{- 1} - 1)) d y \\ = 1 + 2 h \int_{0}^{+ \infty} exp (hpy) {(1 + exp (\frac{- π y}{\sqrt{3}}))}^{- 1} d y \\ = 1 + 2 \int_{1}^{+ \infty} \frac{1}{1 + x^{π / (\sqrt{3} hp)}} d x < + \infty . \end{matrix}$ Case II: Assume δ < ω. For a Lipschitz continuous sample path C_t (γ), we have $\begin{matrix} X_{t} (γ) = & X_{0} e^{- δ t} cos (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{δ X_{0}}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \end{matrix}$ $\begin{matrix} \sqrt{ω^{2} - δ^{2}}) f (s, X_{s} (γ)) d s \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \\ \sqrt{ω^{2} - δ^{2}}) σ (s, X_{s} (γ)) d C_{s} (γ), \end{matrix}$ and $\begin{matrix} Y_{t} (γ) = & Y_{0} e^{- δ t} cos (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{δ Y_{0}}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \\ \sqrt{ω^{2} - δ^{2}}) f (s, X_{s} (γ)) d s \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} e^{- δ (t - s)} sin ((t - s) \\ \sqrt{ω^{2} - δ^{2}}) σ (s, X_{s} (γ)) d C_{s} (γ) . \end{matrix}$ By the strong Lipschitz condition, we have $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | | X_{0} - Y_{0} | e^{- δ t} cos (t \sqrt{ω^{2} - δ^{2}}) \\ + \frac{δ | X_{0} - Y_{0} |}{\sqrt{ω^{2} - δ^{2}}} e^{- δ t} sin (t \sqrt{ω^{2} - δ^{2}}) | \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} | e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) \\ (f (s, X_{s} (γ)) - f (s, Y_{s} (γ))) | d s \\ + \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{t} | e^{- δ (t - s)} sin ((t - s) \sqrt{ω^{2} - δ^{2}}) \\ (σ (s, X_{s} (γ)) - σ (s, Y_{s} (γ))) | | d C_{s} (γ) | \\ \leq | X_{0} - Y_{0} | \end{matrix}$ $\begin{matrix} + \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | f (s, X_{s} (γ)) - f (s, Y_{s} (γ)) | d s \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | σ (s, X_{s} (γ)) - σ (s, Y_{s} (γ)) | d s \\ = | X_{0} - Y_{0} | \end{matrix}$ $\begin{matrix} + \frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{1 s} | (X_{s} (γ) - X_{s} (γ) | d s \\ + \frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{2 s} | (X_{s} (γ) - X_{s} (γ) | d s \\ = | X_{0} - Y_{0} | \\ + \frac{(L_{1 s} + K (γ) L_{2 s})}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} | (X_{s} (γ) - X_{s} (γ) | d s \end{matrix}$ where K (γ) is the Lipschitz constant of C_t (γ). It follows from Gronwall’s inequality that $\begin{matrix} | X_{t} (γ) - Y_{t} (γ) | \\ \leq | X_{0} - Y_{0} | exp (\frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{1 s} d s) \\ exp (\frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{t} L_{2 s} d s) \\ \leq | X_{0} - Y_{0} | exp (\frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{1 s} d s) \\ exp (\frac{K (γ)}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ for any t ≥ 0. Thus we have $\begin{matrix} | X_{t} - Y_{t} | \leq | X_{0} - Y_{0} | exp (\frac{1}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{1 s} d s) \\ exp (\frac{K}{\sqrt{δ^{2} - ω^{2}}} \int_{0}^{+ \infty} L_{2 s} d s) \end{matrix}$ almost surely, where K is a nonnegative uncertain variable such that $M {γ | K (γ) \leq x} \geq 2 {(1 + exp (\frac{- π x}{\sqrt{3}}))}^{- 1} - 1 .$ Taking p-th moment on both sides, we obtain $\begin{matrix} E [| X_{t} - Y_{t} |^{p}] \leq | X_{0} - Y_{0} |^{p} exp (p \int_{0}^{+ \infty} L_{1 s} d s) \\ E [exp (p \frac{K}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{+ \infty} L_{2 s} d s)], \forall t > 0 . \end{matrix}$ Since $\int_{0}^{+ \infty} L_{1 s} d s < + \infty,$ we have $exp (p \int_{0}^{+ \infty} L_{1 s} d s) < + \infty .$ As for $E [exp (p \frac{K}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{+ \infty} L_{2 s} d s)],$ we set $h = \frac{1}{\sqrt{ω^{2} - δ^{2}}} \int_{0}^{+ \infty} L_{2 s} d s < \frac{π}{\sqrt{3} p}$ for simplicity. By the definition of the expected value, we have $\begin{matrix} E [exp (hpK)] \\ = \int_{0}^{+ \infty} M {exp (hpK) \geq x} d x \\ = \int_{0}^{1} M {exp (hpK) \geq x} d x + \\ \int_{1}^{+ \infty} M {exp (hpK) \geq x} d x \\ \leq 1 + \int_{1}^{+ \infty} M {exp (hpK) \geq x} d x \\ = 1 + h \int_{0}^{+ \infty} exp (hpy) M {K \geq y} d y \\ \leq 1 + h \int_{0}^{+ \infty} exp (hpy) (1 - (2 (1 + \\ {exp (\frac{- π y}{\sqrt{3}}))}^{- 1} - 1)) d y \\ = 1 + 2 h \int_{0}^{+ \infty} exp (hpy) {(1 + exp (\frac{- π y}{\sqrt{3}}))}^{- 1} d y \\ = 1 + 2 \int_{1}^{+ \infty} \frac{1}{1 + x^{π / (\sqrt{3} hp)}} d x < + \infty . \end{matrix}$ The theorem is thus proved. □

Remark 3.2. This theorem does not discuss the situation of δ = ω, since resonance phenomenon may occur.

Example 4.2. Consider an uncertain spring vibration equation $\begin{matrix} \frac{d^{2} X_{t}}{d t^{2}} + 20 \frac{d X_{t}}{d t} + 64 X_{t} \\ = exp (- t^{2}) X_{t} d t + exp (- t - X_{t}^{2}) \dot{C_{t}}, \\ t \geq 0, X_{t} \in [l^{s} - l^{e}, l^{l} - l^{e}] . \end{matrix}$ (3) Since f (t, x) = exp(- t²) x and σ (t, x) = exp(- t - x²) satisfy the linear growth condition $| f (t, x) | + | σ (t, x) | \leq | x | + 1, t \geq 0, \forall x \in [l^{s} - l^{e}, l^{l} - l^{e}]$ and satisfy strong Lipschitz condition $| f (t, x) - f (t, y) | \leq exp (- t^{2}) | x - y |,$ $\begin{matrix} | σ (t, x) - σ (t, y) | \leq exp (- t) | x - y |, \\ t \geq 0, \forall x, y \in [l^{s} - l^{e}, l^{l} - l^{e}] \end{matrix}$ with $\int_{0}^{+ \infty} exp (- t^{2}) d t < + \infty,$ $\int_{0}^{+ \infty} exp (- t) d t < \frac{6 π}{\sqrt{3} p}, 0 < p < \frac{π}{\sqrt{3}},$ then uncertain spring vibration equation (3) is stable in p-th moment.

4 Comparison of stability

In this section, we give comparisons of stability in p-th moment and stability in measure of general uncertain spring vibration equation.

Theorem 4.1. If the general uncertain spring vibration equation (1) is stable in p-th moment, then it is stable in measure.

Proof: By Definition 3, for two solutions X_t and Y_t with different initial positions and velocities X₀, Y₀, $\frac{d X_{t}}{d t} |_{t = 0} = 0$ , $\frac{d Y_{t}}{d t} |_{t = 0} = 0$ , we have $lim_{| X_{0} - Y_{0} | \to 0} E [| X_{t} - Y_{t} |^{p}] = 0, \forall t > 0, 0 \leq p \leq + \infty .$ Then for any given real number ε > 0, we have $\begin{matrix} lim_{| X_{0} - Y_{0} | \to 0} M {| X_{t} - Y_{t} | \geq ε} \\ \leq lim_{| X_{0} - Y_{0} | \to 0} \frac{E [| X_{t} - Y_{t} |^{p}]}{ε^{p}} \to 0, \forall t > 0 \end{matrix}$ by Markov inequality, which implies $lim_{| X_{0} - Y_{0} | \to 0} M {| X_{t} - Y_{t} | \leq ε} \to 1 .$ Thus the general uncertain spring vibration equation (1) stable in p-th moment implies the stable in measure. The theorem is completed. □

5 Conclusion

The concept of stability in mean and the concept of stability in measure have been proposed for uncertain spring vibration equations, which calculate the eventual variation in the senses of uncertain measure and expected value, respectively. In contrast, the stability in p-th moment presented in this paper for the uncertain spring vibration differential equations calculates the eventual variation in the senses of p-th moment. Moreover, this paper also discusses the relationships between stability in p-th moment and stability in measure of general uncertain spring vibration equation. The obtained results enrich the stability theory of the uncertain differential equations. For future work, one may consider the extreme value theorems of solution of uncertain spring vibration equation; one may study the first fitting time and time integral of solution of uncertain spring equation.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China Grant Nos. 61873329 and 71471038, Program for Huiyuan Distinguished Young Scholars, UIBE (No. 17JQ09), “the Fundamental Research Funds for the Central Universities” in UIBE (No. CXTD10-05) and Foundation for Disciplinary Development of SITM in UIBE.

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