# 时滞随机控制系统解的估计Estimation of Solutions for Stochastic Control Systems with Time Delays

DOI: 10.12677/AAM.2019.810187, PDF, HTML, XML, 下载: 305  浏览: 435

Abstract: Estimation of solutions for stochastic time delay systems is an important basis for the problem of optimal control systems with time delay. In this paper, we estimate the solution of the state equation of time delay control systems for the general case by using Cauchy-Schwarz and Gronwall inequalities. We use two methods to prove our conclusions, and lay a theoretical foundation for further study of time-delay control problems. And we hope to lay a theoretical foundation for further research on time-delay control.

1. 引言

2. 符号介绍

$\left(\Omega ,F,P\right)$ 是完备的概率空间， ${\left\{{F}_{t}\right\}}_{t\ge 0}$ 是由标准布朗运动 $W\left(t\right)$ 生成的域流。 ${\delta }_{1}$${\delta }_{2}$ 和T是已知常数。 ${L}_{F}^{2}\left(0,T;{R}^{m}\right)$ 表示 ${R}^{m}$${F}_{t}$ 适应过程 $\left\{X\left(t\right),0\le t\le T\right\}$ 的空间，其中 $E\left[{\int }_{s}^{r}{|X\left(t\right)|}^{2}\text{d}t\right]<\infty$$\zeta :\left[-{\delta }_{1},0\right]\to {R}^{n}$ 是连续函数。

$\left\{\begin{array}{l}\text{d}x\left(t\right)=b\left(t,x\left(t\right),x\left(t-{\delta }_{1}\right),u\left(t\right),u\left(t-{\delta }_{2}\right)\right)\text{d}t+\sigma \left(t,x\left(t\right),x\left(t-{\delta }_{1}\right),u\left(t\right),u\left(t-{\delta }_{2}\right)\right)\text{d}W\left(t\right),\\ x\left(t\right)=\zeta \left(t\right),\text{\hspace{0.17em}}t\in \left[-{\delta }_{1},0\right]\end{array}$ (1)

$J\left(u\left(\cdot \right)\right)=E\left\{{\int }_{0}^{T}\left\{L\left(t\right),x\left(t\right),u\left(t\right)\right\}\text{d}t+\Phi \left(x\left(T\right)\right)\right\}$ (2)

(A1) 存在常数 $C>0$ 使得对任意 ${x}_{1}\left(t\right),{x}_{2}\left(t\right),{x}_{1}\left(t-\delta \right),{x}_{2}\left(t-\delta \right),{u}_{1}\left(t\right),{u}_{2}\left(t\right),{u}_{1}\left(t-\delta \right),{u}_{2}\left(t-\delta \right)$ 下面不等式成立：

$\begin{array}{l}{|\rho \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\\ \le C\left[{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}+{|{x}_{1}\left(t-{\delta }_{2}\right)-{x}_{2}\left(t-{\delta }_{2}\right)|}^{2}+{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}+{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\right]\end{array}$

$\Gamma$ 是非空的凸控制域， $U\left[0,T\right]\subset \Gamma$ 表示可容控制集合， $u\left(t\right)$ 定义为

$u\left(t\right)=\left\{\begin{array}{l}0,\text{ }t\in \left[-{\delta }_{2},0\right]\\ u\left(t\right)\in {L}_{F}^{2}\left(0,T;{R}^{m}\right)且u\left(t\right)\in U,\text{ }t\in \left[0,T\right]\end{array}$

3. 主要内容

$\begin{array}{l}E\left[\underset{0\le t\le T}{\mathrm{sup}}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\right]\\ \le C\left\{E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\right\}\end{array}$ (3)

$\begin{array}{l}E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \le E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{1}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\end{array}$

$\begin{array}{l}\le E{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T}{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T}{|{x}_{1}\left(t-{\delta }_{1}\right)-{x}_{2}\left(t-{\delta }_{1}\right)|}^{2}\text{d}t\end{array}$ (4)

$\begin{array}{l}E{\int }_{0}^{T}{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ =E{\int }_{-{\delta }_{2}}^{T-{\delta }_{2}}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t\\ =E{\int }_{-{\delta }_{2}}^{0}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T-{\delta }_{2}}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t\\ \le E{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t\\ =d{\left({u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)}^{2}\end{array}$ (5)

$\begin{array}{l}E{\int }_{0}^{T}{|{x}_{1}\left(t-{\delta }_{2}\right)-{x}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ =E{\int }_{-{\delta }_{2}}^{T-{\delta }_{2}}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t\\ =E{\int }_{-{\delta }_{2}}^{0}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T-{\delta }_{2}}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t\\ \le E{\int }_{0}^{T}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t\end{array}$ (6)

$\begin{array}{l}E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \le E{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T}{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+E{\int }_{0}^{T}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T}{|{x}_{1}\left(t-{\delta }_{2}\right)-{x}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ \le d\left({u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)+E{\int }_{0}^{T}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t\end{array}$ (7)

$\begin{array}{l}E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \le E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{1}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\end{array}$

$\begin{array}{l}\le E{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T}{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t+E{\int }_{0}^{T}{|{x}_{1}\left(t-{\delta }_{2}\right)-{x}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ \le d\left({u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)+E{\int }_{0}^{T}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\text{d}t\end{array}$ (8)

$\begin{array}{l}E\left[\underset{0\le t\le T}{\mathrm{sup}}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\right]\\ \le C\left\{E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\right\}\\ \le C\left\{d{\left({u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)}^{2}+{\int }_{0}^{T}E\left[\underset{0\le t\le \theta }{\mathrm{sup}}{|{x}_{1}\left(t\right)-{x}_{2}\left(t\right)|}^{2}\right]\text{d}\theta \right\}\end{array}$

$\begin{array}{l}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{2}\left(t\right),{x}_{2}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\\ \le C\left[{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}+{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\right]\end{array}$

$\begin{array}{l}E{\int }_{0}^{T}{|b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-b\left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \le CE{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+CE{\int }_{0}^{T}{|{u}_{1}\left(t-{\delta }_{2}\right)-{u}_{2}\left(t-{\delta }_{2}\right)|}^{2}\text{d}t\\ =CE{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+CE{\int }_{-{\delta }_{2}}^{T-{\delta }_{2}}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t\\ =CE{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+CE{\int }_{-{\delta }_{2}}^{0}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+CE{\int }_{0}^{T-{\delta }_{2}}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t\\ \le CE{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t+CE{\int }_{0}^{T}{|{u}_{1}\left(t\right)-{u}_{2}\left(t\right)|}^{2}\text{d}t\\ \le Cd{\left({u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)}^{2}\end{array}$

$\begin{array}{l}E{\int }_{0}^{T}{|\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{1}\left(t\right),{u}_{1}\left(t-{\delta }_{2}\right)\right)-\sigma \left(t,{x}_{1}\left(t\right),{x}_{1}\left(t-{\delta }_{1}\right),{u}_{2}\left(t\right),{u}_{2}\left(t-{\delta }_{2}\right)\right)|}^{2}\text{d}t\\ \le Cd{\left({u}_{1}\left(t\right),{u}_{2}\left(t\right)\right)}^{2}\end{array}$

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