﻿ G-Brown运动驱动的脉冲随机泛函微分方程的指数稳定性 Exponential Stability of Impulsive Stochastic Functional Differential Equations Driven by G-Brownian Motion

Pure Mathematics
Vol. 11  No. 06 ( 2021 ), Article ID: 43357 , 9 pages
10.12677/PM.2021.116135

G-Brown运动驱动的脉冲随机泛函微分方程的指数稳定性

Exponential Stability of Impulsive Stochastic Functional Differential Equations Driven by G-Brownian Motion

Jiping Wang, Guangjie Li*

School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou Guangdong

Received: May 14th, 2021; accepted: Jun. 16th, 2021; published: Jun. 24th, 2021

ABSTRACT

This paper investigates the p-th moment exponential stability of impulsive stochastic functional differential equations driven by G-Brownian motion (G-ISFDEs). By employing the Razumikhin- type method, G-Lyapunov functions, stochastic analysis and algebraic inequality techniques, some sufficient criteria ensuring the p-th moment exponential stability of the trivial solution to G- ISFDEs are established. Meanwhile, an example is presented to illustrate the obtained results.

Keywords:Impulsive Stochastic Functional Differential Equations, G-Brownian Motion, Exponential Stability, Rzumikhin-Type Technique

1. 引言

2. 预备知识

$R=\left(-\infty ,+\infty \right),{R}^{+}=\left[0,+\infty \right)$。对任意的 $x\in {R}^{n},|x|=\sqrt{{x}^{\text{T}}x}$ 表示Euclid范数。若 $A$ 是一个向量或矩阵，

${A}^{\text{T}}$ 代表其转置，且 $‖A‖=\sqrt{{\lambda }_{\mathrm{max}}\left(A{A}^{\text{T}}\right)}$ 表示其范数， $|A|=\sqrt{\text{trace}\left(A{A}^{\text{T}}\right)}$。对 $\forall x,y\in {R}^{n}$$〈x,\text{ }y〉$${x}^{\text{T}}y$

$x,y$ 的内积。 $a\vee b=\mathrm{max}\left\{a,b\right\}$$a\wedge b=\mathrm{min}\left\{a,b\right\}$

$0={t}_{0}<{t}_{1}<{t}_{2}<\cdots <{t}_{N}=T$$\mu \left({\pi }_{T}\right)=\mathrm{max}\left\{|{t}_{i+1}-{t}_{i}|:i=0,1,\cdots ,N-1\right\}$。给定 $p\ge 1$，定义

${M}_{G}^{p,0}\left(\left[0,T\right]\right)=\left\{{\eta }_{t}=\underset{j=0}{\overset{N-1}{\sum }}{\xi }_{j}{I}_{\left[{t}_{j},{t}_{j+1}\right)}\left(t\right):{\xi }_{j}\in {L}_{G}^{p}\left({\Omega }_{{t}_{j}}\right)\right\}$

${M}_{G}^{p}\left(\left[0,T\right]\right)$ 表示 ${M}_{G}^{p,0}\left(\left[0,T\right]\right)$ 在范数 ${‖\eta ‖}_{{M}_{G}^{p,0}\left(\left[0,T\right]\right)}={\left(\frac{1}{T}{\int }_{0}^{T}\stackrel{^}{E}\left[{|{\eta }_{t}|}^{p}\right]\text{d}t\right)}^{1/p}$ 下的完备空间。

$N=\left\{0,1,2,\cdots \right\}$$\phi \left({t}^{+}\right)$$\phi \left({t}^{-}\right)$ 分别表示函数 $\phi \left(t\right)$ 在t时刻的右极限和左极限。令 $\tau >0$

$PC\left(\left[-\tau ,0\right];{R}^{n}\right)=\left\{\phi :\left[-\tau ,0\right]\to {R}^{n}|\phi \left({t}^{+}\right)=\phi \left(t\right),\forall t\in \left[-\tau ,0\right)\right\}$，此外在 $PC\left(\left[-\tau ,0\right];{R}^{n}\right)$ 中，除了有限个点

${\mathcal{L}}_{{\mathcal{F}}_{0}}^{p}\left(\left[-\tau ,0\right];{R}^{n}\right)=\left\{\phi :\phi 是{\mathcal{F}}_{0}\text{-}可测的,P{C}^{b}\left(\left[-\tau ,0\right];{R}^{n}\right)-随机变量,满足\phi \in {M}_{G}^{p}\left(\left[-\tau ,0\right];{R}^{n}\right)\right\}$

${\mathcal{L}}_{{\mathcal{F}}_{t}}^{p}\left(\left[-\tau ,+\infty \right];{R}^{n}\right)=\left\{X:X\in P{C}_{{\mathcal{F}}_{t}}^{p}\left(\left[\Omega ,0\right],{R}^{n}\right),X\text{ }是左极右连续的满足\text{ }X\in {M}_{G}^{p}\left(\left[-\tau ,+\infty \right];{R}^{n}\right)\right\}$

$\left\{\begin{array}{l}\text{d}X\left(t\right)=f\left(t,{X}_{t}\right)\text{d}t+{h}_{ij}\left(t,{X}_{t}\right)\text{d}{〈{B}^{i},{B}^{j}〉}_{t}+{\sigma }_{j}\left(t,{X}_{t}\right)\text{d}{B}_{t}^{j},\text{ }t\ne {t}_{k},\text{ }t\ge {t}_{0},\hfill \\ \Delta X\left({t}_{k}\right)={I}_{k}\left({t}_{k},X\left({t}_{k}^{-}\right)\right),\text{ }t={t}_{k},\text{ }k=1,2,3,\cdots ,\hfill \\ {X}_{{t}_{0}}=\xi ,\hfill \end{array}$ (2.1)

${X}_{t}=X\left(t+\theta \right)\in P{C}_{{\mathcal{F}}_{t}}^{p}\left(\left[-\tau ,0\right],{R}^{n}\right)$${\left\{{〈B,B〉}_{t}\right\}}_{t\ge 0}$ 是G-Brown运动 ${\left\{{B}_{t}\right\}}_{t\ge 0}$ 的二次变差过程。

$f,{h}_{ij},{\sigma }_{j}\in \left[0,T\right]×P{C}_{{\mathcal{F}}_{t}}^{p}\left(\left[-\tau ,0\right],{R}^{n}\right)\to {R}^{n}$ 是Borel-可测的，且对所有的 $T\in {R}^{n}$$X\in {R}^{n}$$f\left(\cdot ,X\right),{h}_{ij}\left(\cdot ,X\right),{\sigma }_{j}\left(\cdot ,X\right)\in {M}_{G}^{p}\left(\left[-\tau ,T\right],{R}^{n}\right)$。脉冲函数 ${I}_{k}\left({t}_{k},X\left({t}_{k}^{-}\right)\right):{R}^{+}×{R}^{n}\to {R}^{n}$ 表示 $X\left(t\right)$${t}_{k}$ 时刻的脉冲扰动。发生脉冲时刻的点 ${t}_{k}$ 满足 $0\le {t}_{0}<{t}_{1}<\cdots <{t}_{k}<\cdots$${t}_{k}\to \infty$ (当 $k\to \infty$ 时， ${t}_{k}\to \infty$ )，

$X\left({t}_{k}^{-}\right)=\underset{h\to {0}^{-}}{\mathrm{lim}}X\left({t}_{k}+h\right)$$X\left({t}_{k}^{+}\right)=\underset{h\to {0}^{+}}{\mathrm{lim}}X\left({t}_{k}+h\right)$$\Delta X\left({t}_{k}\right)=X\left({t}_{k}\right)-X\left({t}_{k}^{-}\right)$

${\int }_{0}^{t}{h}_{ij}\left(s,{X}_{s}\right)\text{d}{〈{B}^{i},{B}^{j}〉}_{s}:=\underset{i,j=1}{\overset{d}{\sum }}{\int }_{0}^{t}{h}_{ij}\left(s,{x}_{s}\right)\text{d}{〈{B}^{i},{B}^{j}〉}_{s},$

${\int }_{0}^{t}{\sigma }_{j}\left(s,{x}_{s}\right)\text{d}{B}_{s}^{j}:=\underset{j=1}{\overset{d}{\sum }}{\int }_{0}^{t}{\sigma }_{j}\left(s,{x}_{s}\right)\text{d}{B}_{s}^{j}.$

$\stackrel{^}{E}{|X\left(t;{t}_{0},\xi \right)|}^{p}\le C\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left(t-{t}_{0}\right)},\text{ }t\ge {t}_{0},$

${V}_{t}\left(t,X\right)=\frac{\partial V\left(t,X\right)}{\partial t}$${V}_{X}\left(t,X\right)=\left(\frac{\partial V\left(t,X\right)}{\partial {X}_{1}},\frac{\partial V\left(t,X\right)}{\partial {X}_{2}},\cdots ,\frac{\partial V\left(t,X\right)}{\partial {X}_{n}}\right)$${V}_{XX}\left(t,X\right)={\left(\frac{{\partial }^{2}V\left(t,X\right)}{\partial {X}_{i}\partial {X}_{j}}\right)}_{n×n}$ 对每一个

$V\in {C}^{1,2}\left(\left[{t}_{0}-\tau ,\infty \right)×{R}^{n},{R}^{+}\right)$，定义算子：

$\begin{array}{c}LV\left(t,{X}_{t}\right)={V}_{t}\left(t,X\right)+〈{V}_{X}\left(t,X\right),f\left(t,{X}_{t}\right)〉\\ \text{\hspace{0.17em}}\text{ }+G\left(〈{V}_{X}\left(t,X\right),h\left(t,{X}_{t}\right)〉+〈{V}_{XX}\left(t,X\right)\sigma \left(t,{X}_{t}\right),\sigma \left(t,{X}_{t}\right)〉\right),\end{array}$

$\begin{array}{l}〈{V}_{X}\left(t,X\right),h\left(t,{X}_{t}\right)〉+〈{V}_{XX}\left(t,X\right)\sigma \left(t,{X}_{t}\right),\sigma \left(t,{X}_{t}\right)〉\\ :={\left[〈{V}_{X}\left(t,X\right),{h}_{ij}\left(t,{X}_{t}\right)+{h}_{ji}\left(t,{X}_{t}\right)〉+〈{V}_{XX}\left(t,X\right){\sigma }_{i}\left(t,{X}_{t}\right),{\sigma }_{j}\left(t,{X}_{t}\right)〉\right]}_{i,j=1}^{d}.\end{array}$

3. 主要结果

(i) 对 $\forall \left(t,X\right)\in \left[{t}_{0}-\tau ,\infty \right)×{R}^{n}$${C}_{1}{|X|}^{p}\le V\left(t,X\right)\le {C}_{2}{|X|}^{p}$

(ii) 对所有的 $k\in N$$X\in {\mathcal{L}}_{{\mathcal{F}}_{t}}^{p}\left(\left[-\tau ,+\infty \right];{R}^{n}\right)$

$V\left({t}_{k},X+{I}_{k}\left({t}_{k},X\right)\right)\le {d}_{k}V\left({t}_{k}^{-},X\right),$

(iii) 对所有的 $t\ge {t}_{0}$$t\ne {t}_{k},k\in \left\{1,2,\cdots \right\}$$\phi \in {\mathcal{L}}_{{\mathcal{F}}_{0}}^{p}\left(\left[-\tau ,0\right];{R}^{n}\right)$

$\stackrel{^}{E}LV\left(t,\phi \right)\le \left(-\lambda +{\lambda }_{1}\left(t\right)\right)\stackrel{^}{E}V\left(t,\phi \left(0\right)\right),$

$\stackrel{^}{E}LV\left(t,\phi \right)\le \left(-\lambda +{\lambda }_{1}\left(t\right)\right)\stackrel{^}{E}V\left(t,\phi \left(0\right)\right),$

$\theta \in \left[-\tau ,0\right]$$\stackrel{^}{E}V\left(t+\theta ,\phi \right) (这里 $q\ge \gamma {\text{e}}^{\lambda \tau }$ )， $\gamma ={\mathrm{max}}_{k\in \left\{1,2,\cdots \right\}}\left\{\frac{1}{{d}_{k}}\right\}$${\lambda }_{1}\left(t\right):\left[{t}_{0},\infty \right)\to R$${\lambda }_{1}\left(t\right)$$\left[{t}_{k},{t}_{k+1}\right)$ 上是连续的，对所有的 $k\in \left\{1,2,\cdots \right\}$${\mathrm{lim}}_{t-{t}_{k}^{-}}{\lambda }_{1}\left(t\right)=\lambda \left({t}_{k}^{-}\right)$，且 ${\int }_{{t}_{0}}^{+\infty }{\lambda }_{1}^{+}\left(s\right)\text{d}s<\infty$ (这里

${\lambda }_{1}^{+}\left(s\right)=\mathrm{max}\left\{{\lambda }_{1}\left(s\right),0\right\}$ )，则方程(2.1)的平凡解是p-阶矩指数稳定的。

$\stackrel{^}{E}V\left(t,X\left(t\right)\right)\le {C}_{2}\stackrel{^}{E}{|X\left(t\right)|}^{p}\le {C}_{2}\stackrel{^}{E}{‖\xi ‖}^{p}\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)},t\in \left[{t}_{0}-\tau ,{t}_{0}\right).$ (3.1)

$\stackrel{^}{E}V\left(t,X\left(t\right)\right)\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{k}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s},t\in \left[{t}_{k},{t}_{k+1}\right),\text{\hspace{0.17em}}k\in N.$ (3.2)

$\stackrel{^}{E}V\left(t,X\left(t\right)\right)\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s},\text{\hspace{0.17em}}t\in \left[{t}_{0},{t}_{1}\right),$ (3.3)

$\stackrel{^}{E}V\left(t,X\left(t\right)\right)>M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s}.$

${t}_{1}^{\ast }=\mathrm{inf}\left\{t\in \left[{t}_{0},{t}_{1}\right):\stackrel{^}{E}V\left(t,X\left(t\right)\right)>M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\right\}$。注意 $\stackrel{^}{E}V\left(t,X\left(t\right)\right)$$\left[{t}_{0},{t}_{1}\right)$ 是连续的，故

${t}_{1}^{\ast }\in \left[{t}_{0},{t}_{1}\right)$

$EV\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)=M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{1}^{\ast }}{\lambda }_{1}^{+}\left(s\right)\text{d}s}$ (3.4)

$\stackrel{^}{E}V\left(t,X\left(t\right)\right) (3.5)

$\stackrel{^}{E}V\left({t}_{n},X\left({t}_{n}\right)\right)>M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{n}}{\lambda }_{1}^{+}\left(s\right)\text{d}s},\text{\hspace{0.17em}}{t}_{n}\in \left[{t}_{1}^{\ast },{t}_{1}\right),$ (3.6)

$\begin{array}{c}\stackrel{^}{E}V\left({t}_{1}^{\ast }+\theta ,X\left({t}_{1}^{\ast }+\theta \right)\right)\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{1}^{\ast }+\theta }{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{1}^{\ast }}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ (3.7)

$\stackrel{^}{E}LV\left({t}_{1}^{\ast },{X}_{{t}_{1}^{\ast }}\right)\le \left(-\lambda +{\lambda }_{1}\left({t}_{1}^{\ast }\right)\right)\stackrel{^}{E}V\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)$ (3.8)

$\stackrel{^}{E}LV\left(t,{X}_{t}\right)\le \left(-\lambda +{\lambda }_{1}\left(t\right)\right)\stackrel{^}{E}V\left(t,X\left(t\right)\right),\text{\hspace{0.17em}}t\in \left[{t}_{1}^{\ast },{t}_{1}^{\ast }+h\right).$ (3.9)

$\begin{array}{l}\text{d}\left({\text{e}}^{\lambda t}V\left(t,X\left(t\right)\right)\right)\\ ={\text{e}}^{\lambda t}\left[\lambda V\left(t,X\left(t\right)\right)+{V}_{t}\left(t,X\left(t\right)\right)+〈{V}_{X}\left(t,X\left(t\right)\right),f\left(t,{X}_{t}\right)〉\right]\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\text{e}}^{\lambda t}\left[〈{V}_{X}\left(t,X\left(t\right)\right),{h}_{ij}\left(t,{X}_{t}\right)〉+\frac{1}{2}〈{V}_{XX}\left(t,X\left(t\right)\right){\sigma }_{i}\left(t,{X}_{t}\right),{\sigma }_{j}\left(t,{X}_{t}\right)〉\right]\text{d}{〈{B}^{i},{B}^{j}〉}_{t}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\text{e}}^{\lambda t}〈{V}_{X}\left(t,X\left(t\right)\right),{\sigma }_{j}\left(t,{X}_{t}\right)〉\text{d}{B}_{t}^{j},\end{array}$

$\begin{array}{c}{\text{e}}^{\lambda t}V\left(t,X\left(t\right)\right)={\text{e}}^{\lambda {t}_{1}^{\ast }}V\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)+{\int }_{{t}_{1}^{\ast }}^{t}{\text{e}}^{\lambda s}\left(\lambda V\left(s,X\left(s\right)\right)+LV\left(s,{X}_{s}\right)\right)\text{d}s\\ \text{\hspace{0.17em}}\text{ }+{M}_{t}^{{t}_{1}^{\ast }}+{\int }_{{t}_{1}^{\ast }}^{t}{\text{e}}^{\lambda s}〈{V}_{X}\left(s,X\left(s\right)\right),\sigma \left(s,{X}_{s}\right)〉\text{d}{B}_{s},\end{array}$ (3.10)

$\begin{array}{c}{M}_{t}^{{t}_{1}^{*}}={\int }_{s}^{t}\left[〈{V}_{X}\left(s,X\left(s\right)\right),{h}_{ij}\left(s,{X}_{s}\right)〉+\frac{1}{2}〈{V}_{XX}\left(s,X\left(s\right)\right){\sigma }_{i}\left(s,{X}_{s}\right),{\sigma }_{j}\left(s,{X}_{s}\right)〉\right]\text{d}{〈{B}^{i},{B}^{j}〉}_{s}\\ \text{\hspace{0.17em}}\text{ }-{\int }_{s}^{t}G\left(〈{V}_{X}\left(s,X\left(s\right)\right),h\left(s,{X}_{s}\right)〉+〈{V}_{XX}\left(s,X\left(s\right)\right)\sigma \left(s,{X}_{s}\right),\sigma \left(s,{X}_{s}\right)〉\right)\text{d}s.\end{array}$ (3.11)

$\begin{array}{c}\stackrel{^}{E}\left[{\text{e}}^{\lambda t}V\left(t,X\left(t\right)\right)\right]=\stackrel{^}{E}\left[{\text{e}}^{\lambda {t}_{1}^{\ast }}V\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)\right]+\stackrel{^}{E}\left[{\int }_{{t}_{1}^{\ast }}^{t}{\text{e}}^{\lambda s}\left(\lambda V\left(s,X\left(s\right)\right)+LV\left(s,{X}_{s}\right)\right)\right]\text{d}s\\ \le \stackrel{^}{E}\left[{\text{e}}^{\lambda {t}_{1}^{\ast }}V\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)\right]+{\int }_{{t}_{1}^{\ast }}^{t}{\lambda }_{1}\left(s\right)\stackrel{^}{E}\left({\text{e}}^{\lambda s}V\left(s,X\left(s\right)\right)\right)\text{d}s.\end{array}$ (3.12)

$\stackrel{^}{E}\left[{\text{e}}^{\lambda t}V\left(t,X\left(t\right)\right)\right]\le \stackrel{^}{E}\left[{\text{e}}^{\lambda {t}_{1}^{\ast }}V\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)\right]{\text{e}}^{{\int }_{{t}_{1}^{\ast }}^{t}{\lambda }_{1}\left(s\right)\text{d}s},$

$\stackrel{^}{E}\left[V\left(t,X\left(t\right)\right)\right]\le {\text{e}}^{-\lambda \left(t-{t}_{1}^{\ast }\right)}\stackrel{^}{E}\left[V\left({t}_{1}^{\ast },X\left({t}_{1}^{\ast }\right)\right)\right]{\text{e}}^{{\int }_{{t}_{1}^{\ast }}^{t}{\lambda }_{1}\left(s\right)\text{d}s},$ (3.13)

$\begin{array}{c}\stackrel{^}{E}\left[V\left({t}_{1}^{\ast }+h,X\left({t}_{1}^{\ast }+h\right)\right)\right]\le {\text{e}}^{-\lambda h}\stackrel{^}{E}\left[V\left({t}_{1}^{*},X\left({t}_{1}^{*}\right)\right)\right]{\text{e}}^{{\int }_{{t}_{1}^{\ast }}^{t}{\lambda }_{1}\left(s\right)\text{d}s}\\ \le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}+h-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ (3.14)

$\stackrel{^}{E}\left[V\left(t,X\left(t\right)\right)\right]\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s},\text{\hspace{0.17em}}t\in \left[{t}_{0},{t}_{1}\right)$

$E\left[V\left(t,X\left(t\right)\right)\right]\le ME{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{k}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s},\text{\hspace{0.17em}}t\in \left[{t}_{k-1},{t}_{k}\right)$ (3.15)

$\stackrel{^}{E}\left[V\left(t,X\left(t\right)\right)\right]\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s},\text{\hspace{0.17em}}t\in \left[{t}_{m},{t}_{m+1}\right),$ (3.16)

${t}_{2}^{\ast }=\mathrm{inf}\left\{t\in \left[{t}_{m},{t}_{m+1}\right):\stackrel{^}{E}\left[V\left(t,X\left(t\right)\right)\right]>M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\right\}.$

$\begin{array}{c}\stackrel{^}{E}V\left({t}_{m},X\left({t}_{m}\right)\right)\le {d}_{m}\stackrel{^}{E}V\left({t}_{m}^{-},X\left({t}_{m}^{-}\right)\right)\le {d}_{m}M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m}-{t}_{0}\right)}\\ \le {\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{m}\right)}M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{m}}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ =M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{m}}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\end{array}$ (3.17)

$\stackrel{^}{E}\left[V\left({t}_{2}^{\ast },X\left({t}_{2}^{\ast }\right)\right)\right]=M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{\ast }}{\lambda }_{1}^{+}\left(s\right)\text{d}s},$ (3.18)

$\stackrel{^}{E}\left[V\left(t,X\left(t\right)\right)\right] (3.19)

$\stackrel{^}{E}V\left({t}_{n},X\right)\left({t}_{n}\right)>M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{n}}{\lambda }_{1}^{+}\left(s\right)\text{d}s},\text{\hspace{0.17em}}{t}_{n}\in \left[{t}_{2}^{\ast },{t}_{m+1}\right).$ (3.20)

$-\tau \le \theta \le 0$，存在一个整数 $j\in \left[0,k\right]\text{\hspace{0.17em}}\left(k\le m\right)$ 满足 ${t}_{2}^{\ast }+\theta \in \left[{t}_{j},{t}_{j+1}\right)$。从而

$\begin{array}{c}\stackrel{^}{E}V\left({t}_{2}^{\ast }+\theta ,X\left({t}_{2}^{\ast }+\theta \right)\right)\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{j+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{\ast }+\theta }{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{2}^{\ast }+\theta -{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{\ast }}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}+{t}_{2}^{\ast }-{t}_{m+1}\right)}{\text{e}}^{-\lambda \theta }{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{\ast }}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{\lambda \left({t}_{m+1}-{t}_{2}^{\ast }\right)}{\text{e}}^{\lambda \tau }{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{*}}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\end{array}$

$\begin{array}{l}\le M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{\lambda \left({t}_{m+1}-{t}_{m}\right)}{\text{e}}^{\lambda \tau }{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{*}}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le \gamma {\text{e}}^{\lambda \tau }M\stackrel{^}{E}{‖\xi ‖}^{p}{\text{e}}^{-\lambda \left({t}_{m+1}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{{t}_{2}^{\ast }}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le q\stackrel{^}{E}V\left({t}_{2}^{\ast },X\left({t}_{2}^{\ast }\right)\right).\end{array}$ (3.21)

$\stackrel{^}{E}LV\left({t}_{2}^{\ast },{X}_{{t}_{2}^{*}}\right)\le \left(-\lambda +{\lambda }_{1}\left({t}_{2}^{*}\right)\right)\stackrel{^}{E}V\left({t}_{2}^{*},X\left({t}_{2}^{*}\right)\right).$ (3.22)

$\begin{array}{c}\stackrel{^}{E}{|X\left(t\right)|}^{p}\le \frac{M\stackrel{^}{E}{‖\xi ‖}^{p}}{{C}_{1}}{\text{e}}^{-\lambda \left({t}_{k}-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le \frac{M\stackrel{^}{E}{‖\xi ‖}^{p}}{{C}_{1}}{\text{e}}^{-\lambda \left(t-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{t}{\lambda }_{1}^{+}\left(s\right)\text{d}s}\\ \le \frac{M\stackrel{^}{E}{‖\xi ‖}^{p}}{{C}_{1}}{\text{e}}^{-\lambda \left(t-{t}_{0}\right)}{\text{e}}^{{\int }_{{t}_{0}}^{+\infty }{\lambda }_{1}^{+}\left(s\right)\text{d}s}.\end{array}$

$\stackrel{^}{E}{|X\left(t\right)|}^{p}\le \frac{CM\stackrel{^}{E}{‖\xi ‖}^{p}}{{C}_{1}}{\text{e}}^{-\lambda \left(t-{t}_{0}\right)},\text{\hspace{0.17em}}t\ge {t}_{0},$ (3.23)

4. 例子

$\left\{\begin{array}{l}\text{d}X\left(t\right)=-3X\left(t\right)\text{d}t+X\left(t\right)\text{d}{〈B,B〉}_{t}+\frac{|\mathrm{sin}t|}{\sqrt{1+{t}^{2}}}X\left(t-|\mathrm{sin}t|\right)\text{d}{B}_{t},\text{\hspace{0.17em}}t\ge {t}_{0},t\ne {t}_{k},\hfill \\ \Delta X\left(t\right)=\frac{1}{{k}^{2}}X\left({t}_{k},X\left({t}_{k}\right)\right),\text{\hspace{0.17em}}t={t}_{k},\text{\hspace{0.17em}}k=1,2,\cdots .\hfill \end{array}$

$p=2$$V\left(t,X\left(t\right)\right)={|X\left(t\right)|}^{2}$$f\left(t,X\left(t\right)\right)=-3{|X\left(t\right)|}^{2}$$h\left(t,X\left(t\right)\right)=X\left(t\right)$

$\sigma \left(t,X\left(t-\tau \left(t\right)\right)\right)=\frac{|\mathrm{sin}t|}{\sqrt{1+{t}^{2}}}X\left(t-|\mathrm{sin}t|\right)$。计算得

$\begin{array}{c}LV\left(t,{X}_{t}\right)={V}_{t}\left(t,X\left(t\right)\right)+〈{V}_{X}\left(t,X\left(t\right)\right),f\left(t,X\left(t\right)\right)〉\\ \text{ }\text{\hspace{0.17em}}+G\left(〈{V}_{X}\left(t,X\left(t\right)\right),h\left(t,X\left(t\right)\right)+h\left(t,X\left(t\right)\right)〉\\ \text{\hspace{0.17em}}\text{ }+〈{V}_{XX}\left(t,X\left(t\right)\right)\sigma \left(t,X\left(t-\tau \left(t\right)\right)\right),\sigma \left(t,X\left(t-\tau \left(t\right)\right)\right)〉\right)\\ =-6{|X\left(t\right)|}^{2}+G\left(4{|X\left(t\right)|}^{2}+2\frac{{\mathrm{sin}}^{2}t}{1+{t}^{2}}{X}^{2}\left(t-|\mathrm{sin}t|\right)\right)\\ \le -6{|X\left(t\right)|}^{2}+2{|X\left(t\right)|}^{2}+\frac{{\mathrm{sin}}^{2}t}{1+{t}^{2}}{X}^{2}\left(t-|\mathrm{sin}t|\right).\end{array}$

${t}_{k+1}-{t}_{k}=\frac{1}{2}$${d}_{k}={\text{e}}^{-2}$$k=1,2,\cdots$$\gamma ={\text{e}}^{3}$$q={\text{e}}^{7}$$\tau \left(t\right)=|\mathrm{sin}t|$，进一步可得

$\stackrel{^}{E}LV\left(t,{X}_{t}\right)\le \left(-4+{\text{e}}^{7}\frac{{\mathrm{sin}}^{2}t}{1+{t}^{2}}\right)\stackrel{^}{E}{|X\left(t\right)|}^{2},$

Exponential Stability of Impulsive Stochastic Functional Differential Equations Driven by G-Brownian Motion[J]. 理论数学, 2021, 11(06): 1221-1229. https://doi.org/10.12677/PM.2021.116135

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21. NOTES

*通讯作者。