无穷维随机系统的稳定性、可控性及其应用
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摘要
现实世界中的任何系统都存在随机因素,为了更准确的描述实际系统,设计更好的控制方案,在建立实际系统的模型时就必须充分考虑随机因素的影响–建立随机系统模型.无穷维随机系统是用随机偏微分方程、随机积分方程或抽象空间中的随机发展方程来描述的一类随机系统,主要特点是包含有随机噪声且状态变量属于无穷维的函数空间.目前,无穷维随机系统模型已经广泛地应用于现代量子力学、流体力学、大气海洋预报、图像处理、工业控制、经济、生物学等领域,关于无穷维随机系统的研究已经成为现代控制理论和数学上的研究热点.
本文基于无穷维随机分析、线性算子半群理论、现代偏微分方程的基本理论,利用Lyapunov方法、Banach不动点定理、线性算子不等式等方法,系统的研究了无穷维随机系统的稳定性和可控性问题,提出了针对无穷维随机系统新的分析方法,获得了若干具有重要意义的理论和应用结果.本学位论文的主要工作有以下几个方面:
1、对无穷维随机系统及其稳定性和可控性的研究进展、研究方法进行了概述,并简单介绍了关于无穷维空间的随机积分及其常用性质和重要不等式,列出了无穷维随机偏微分方程的Ito|^公式.
2、对在非Lipschitz条件下的非线性漂移系数和扩散系数的随机双曲方程解的存在唯一性进行了讨论.利用适度解的Green函数表示,通过构造解的逼近列,在适当的空间证明其收敛到对应的适度解,得到了具有广泛代表意义的随机双曲方程适度解的存在唯一性.并且,所得结果包含了非线性系数满足通常的Lipschitz条件和线性增长条件的情形.
3、利用经典的Banach不动点定理分析了具有脉冲影响的无穷维半线性随机系统的完全可控性问题.通过引入适当的反馈控制与可控性算子,利用无穷维脉冲随机微分方程适度解的常数变易公式、无穷维随机分析的B-D-G不等式,证明了系统在具有脉冲激励条件下,在适当小的时间段内,系统仍然是完全可控的,并给出具体例子说明其应用.
4、对无界区域上一阶非线性随机双曲系统的稳定性进行了分析.利用傅里叶变换、C0半群理论,在无穷小生成元满足强双曲条件下,建立了适度解的均方指数稳定和几乎必然指数稳定的充分条件.并且,给出一个实际例子说明其应用.
5、对作用在时滞项是有界算子的无穷维随机线性时滞系统的稳定性进行了分析.利用一种新的工具–线性算子不等式(Linear Operator Inequality, LOI),对抽象的无穷维线性随机时滞系统,建立了系统状态按照给定速率衰减的均方指数稳定的充分条件.这里时滞既可以是常时滞,也可以是变时滞,或多时滞、多变时滞.并且指出,无穷维情形下的算子不等式可以看作是有限维线性矩阵不等式的推广.
6、基于线性算子不等式对随机时滞热传导方程、随机时滞波动方程的稳定性问题进行了研究.对常系数随机时滞热传导方程,借助偏微分方程的Green公式、Poincare′不等式,分析了时滞和噪声对系统均方指数稳定性的影响.而对一维常系数线性随机时滞波动方程,则由于双曲系统具有能量守恒性,研究了系统存在耗散项的情况下,构造适当的Lyapunov函数,得到稳定性结果,以线性矩阵不等式的形式给出了强解均方指数稳定的充分条件.
7.对随机半线性抛物方程的边界镇定和H∞问题进行了研究.通过设计边界静态反馈控制律,以线性矩阵不等式(LMI)的形式分别给出了使得系统均方指数镇定和鲁棒镇定的充分条件.
最后总结全文并指出进一步可研究的方向.
Stochastic perturbation is inevitable in any real world system. To describe the real systemsmore exactly and further to design better controllers, it is necessary to take account of stochasticfactors. Infinite dimensional stochastic systems are described by stochastic partial differentialequations, stochastic integral equations or stochastic evolution equations in abstract spaces. Itsmain characters are being in?uenced by stochastic noises and the state variables belonging toinfinite dimensional function space. Nowadays, infinite dimensional stochastic systems havebeen widely applied in many fields, such as modern quantum mechanics, hydromechanics,ocean atmosphere forecasting, image processing, industrial control, economics, biology, etc.Infinite dimensional stochastic systems and their control theory have become a hot spot inmodern control theory and mathematics.
Based on the theory of infinite dimensional stochastic analysis, linear operator semigroup,and modern partial differential equations, this dissertation explores some new analysis tech-nique and systematically studies the stability and controllability of infinite dimensional stochas-tic systems by means of Lyapunov method, Banach fixed point theorem and linear operatorinequality. Some important theoretical and practical results are obtained. The main contentsand contribution of this dissertation are summarized as follows:
1. An introduction to the latest progress and research method in stability and controlla-bility of infinite dimensional stochastic systems is given. Also, stochastic integral in infinitedimensional spaces and its properties, some important inequalities are presented. And then theIt(o|^) formula for infinite dimensional stochastic partial differential equations is given.
2. The existence and uniqueness of the solution of stochastic hyperbolic equation withnon-Lipschitz drift and diffuse coefficients are discussed. By using Green function presentationof mild solution and the formula for the variation of parameters, approximate sequence of mildsolution are constructed in some suitable space and proven converging to the real solution.The obtained existence and uniqueness theorem is typical. It is easy to find that the Lipschitzcondition is only a special case of our result.
3. The complete controllability problem of semilinear impulsive stochastic systems ininfinite dimensional space is concerned by using Banach fixed point theorems. Utilizing theformula for the variation of parameters, B-D-G inequality and introducing feedback control,the systems are proven to be completely controllable in certain small interval. An example isgiven to illustrate the theorem.
4.Stability of first order nonlinear stochastic hyperbolic systems in whole space is con-cerned. Employing Fourier transformation and C0 semigroup theory, sufficient conditions en-suring exponentially stable in mean square and almost surely exponentially stable are givenunder strong hyperbolic assumption. An example is provided to illustrate our theory.
5. The stability of a class of linear stochastic delay systems in infinite dimensional space isconsidered. The system delay is admitted to be unknown, time-varying, multi delays and multivarying delays, but the operator acting on the delayed states is bounded. Sufficient conditionsensuring exponential stability of abstract infinite dimensional stochastic time delay systems atgiven decaying rate are derived in the form of LOI. We point out the LOI in infinite dimensionalspaces is a generalization of LMI in finite dimensional spaces.
6. Based on linear operator inequality, the stability of stochastic heat equations andstochastic wave equations with time delay is considered. For stochastic delay heat equationswith constant coefficients, the in?uence of delay and noise to the mean square exponential sta-bility is discussed by Green formula and Poincare′inequality. Noting the conversation of energy,stability of one order linear stochastic delay wave equation with dissipation term is considered.By constructing Lyapunov functions, sufficient conditions ensuring mean square exponentialstability of strong solution is obtained in the form of LMI.
7. The boundary stabilization and robust H_∞problem of semilinear stochastic parabolicequations are considered. By designing boundary static state controller, sufficient conditions ofmean square exponential stability and robust stabilizations are derived in the form of LMI.
Finally, the main results of the dissertation are summarized, and the issues of future inves-tigation are proposed.
本文基于无穷维随机分析、线性算子半群理论、现代偏微分方程的基本理论,利用Lyapunov方法、Banach不动点定理、线性算子不等式等方法,系统的研究了无穷维随机系统的稳定性和可控性问题,提出了针对无穷维随机系统新的分析方法,获得了若干具有重要意义的理论和应用结果.本学位论文的主要工作有以下几个方面:
1、对无穷维随机系统及其稳定性和可控性的研究进展、研究方法进行了概述,并简单介绍了关于无穷维空间的随机积分及其常用性质和重要不等式,列出了无穷维随机偏微分方程的Ito|^公式.
2、对在非Lipschitz条件下的非线性漂移系数和扩散系数的随机双曲方程解的存在唯一性进行了讨论.利用适度解的Green函数表示,通过构造解的逼近列,在适当的空间证明其收敛到对应的适度解,得到了具有广泛代表意义的随机双曲方程适度解的存在唯一性.并且,所得结果包含了非线性系数满足通常的Lipschitz条件和线性增长条件的情形.
3、利用经典的Banach不动点定理分析了具有脉冲影响的无穷维半线性随机系统的完全可控性问题.通过引入适当的反馈控制与可控性算子,利用无穷维脉冲随机微分方程适度解的常数变易公式、无穷维随机分析的B-D-G不等式,证明了系统在具有脉冲激励条件下,在适当小的时间段内,系统仍然是完全可控的,并给出具体例子说明其应用.
4、对无界区域上一阶非线性随机双曲系统的稳定性进行了分析.利用傅里叶变换、C0半群理论,在无穷小生成元满足强双曲条件下,建立了适度解的均方指数稳定和几乎必然指数稳定的充分条件.并且,给出一个实际例子说明其应用.
5、对作用在时滞项是有界算子的无穷维随机线性时滞系统的稳定性进行了分析.利用一种新的工具–线性算子不等式(Linear Operator Inequality, LOI),对抽象的无穷维线性随机时滞系统,建立了系统状态按照给定速率衰减的均方指数稳定的充分条件.这里时滞既可以是常时滞,也可以是变时滞,或多时滞、多变时滞.并且指出,无穷维情形下的算子不等式可以看作是有限维线性矩阵不等式的推广.
6、基于线性算子不等式对随机时滞热传导方程、随机时滞波动方程的稳定性问题进行了研究.对常系数随机时滞热传导方程,借助偏微分方程的Green公式、Poincare′不等式,分析了时滞和噪声对系统均方指数稳定性的影响.而对一维常系数线性随机时滞波动方程,则由于双曲系统具有能量守恒性,研究了系统存在耗散项的情况下,构造适当的Lyapunov函数,得到稳定性结果,以线性矩阵不等式的形式给出了强解均方指数稳定的充分条件.
7.对随机半线性抛物方程的边界镇定和H∞问题进行了研究.通过设计边界静态反馈控制律,以线性矩阵不等式(LMI)的形式分别给出了使得系统均方指数镇定和鲁棒镇定的充分条件.
最后总结全文并指出进一步可研究的方向.
Stochastic perturbation is inevitable in any real world system. To describe the real systemsmore exactly and further to design better controllers, it is necessary to take account of stochasticfactors. Infinite dimensional stochastic systems are described by stochastic partial differentialequations, stochastic integral equations or stochastic evolution equations in abstract spaces. Itsmain characters are being in?uenced by stochastic noises and the state variables belonging toinfinite dimensional function space. Nowadays, infinite dimensional stochastic systems havebeen widely applied in many fields, such as modern quantum mechanics, hydromechanics,ocean atmosphere forecasting, image processing, industrial control, economics, biology, etc.Infinite dimensional stochastic systems and their control theory have become a hot spot inmodern control theory and mathematics.
Based on the theory of infinite dimensional stochastic analysis, linear operator semigroup,and modern partial differential equations, this dissertation explores some new analysis tech-nique and systematically studies the stability and controllability of infinite dimensional stochas-tic systems by means of Lyapunov method, Banach fixed point theorem and linear operatorinequality. Some important theoretical and practical results are obtained. The main contentsand contribution of this dissertation are summarized as follows:
1. An introduction to the latest progress and research method in stability and controlla-bility of infinite dimensional stochastic systems is given. Also, stochastic integral in infinitedimensional spaces and its properties, some important inequalities are presented. And then theIt(o|^) formula for infinite dimensional stochastic partial differential equations is given.
2. The existence and uniqueness of the solution of stochastic hyperbolic equation withnon-Lipschitz drift and diffuse coefficients are discussed. By using Green function presentationof mild solution and the formula for the variation of parameters, approximate sequence of mildsolution are constructed in some suitable space and proven converging to the real solution.The obtained existence and uniqueness theorem is typical. It is easy to find that the Lipschitzcondition is only a special case of our result.
3. The complete controllability problem of semilinear impulsive stochastic systems ininfinite dimensional space is concerned by using Banach fixed point theorems. Utilizing theformula for the variation of parameters, B-D-G inequality and introducing feedback control,the systems are proven to be completely controllable in certain small interval. An example isgiven to illustrate the theorem.
4.Stability of first order nonlinear stochastic hyperbolic systems in whole space is con-cerned. Employing Fourier transformation and C0 semigroup theory, sufficient conditions en-suring exponentially stable in mean square and almost surely exponentially stable are givenunder strong hyperbolic assumption. An example is provided to illustrate our theory.
5. The stability of a class of linear stochastic delay systems in infinite dimensional space isconsidered. The system delay is admitted to be unknown, time-varying, multi delays and multivarying delays, but the operator acting on the delayed states is bounded. Sufficient conditionsensuring exponential stability of abstract infinite dimensional stochastic time delay systems atgiven decaying rate are derived in the form of LOI. We point out the LOI in infinite dimensionalspaces is a generalization of LMI in finite dimensional spaces.
6. Based on linear operator inequality, the stability of stochastic heat equations andstochastic wave equations with time delay is considered. For stochastic delay heat equationswith constant coefficients, the in?uence of delay and noise to the mean square exponential sta-bility is discussed by Green formula and Poincare′inequality. Noting the conversation of energy,stability of one order linear stochastic delay wave equation with dissipation term is considered.By constructing Lyapunov functions, sufficient conditions ensuring mean square exponentialstability of strong solution is obtained in the form of LMI.
7. The boundary stabilization and robust H_∞problem of semilinear stochastic parabolicequations are considered. By designing boundary static state controller, sufficient conditions ofmean square exponential stability and robust stabilizations are derived in the form of LMI.
Finally, the main results of the dissertation are summarized, and the issues of future inves-tigation are proposed.
引文
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