随机时滞微分方程的稳定性研究
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摘要
随机时滞微分方程作为一种重要的数学模型,可以把它看作是既考虑了确定性时滞微分方程模型问题又考虑了随机因素,也可以视为既考虑了非确定性随机微分方程模型问题又考虑了时滞因素的影响,所以随机时滞微分方程通常能更真实地模拟科学实际中的问题,也因此被广泛地应用到控制论、神经网络、生物学、金融学、化学反应工程等众多领域。由于随机扰动和时滞通常会导致系统不稳定,因此研究随机时滞微分系统的稳定性是很有必要的。
简单介绍随机时滞微分方程稳定性研究背景和研究现状,给出了本文中常用的基本概念、基础知识和重要引理。
首先研究具有变时滞线性中立型随机微分方程的指数p稳定性以及具有混合变时滞线性中立型随机微分方程的渐近p稳定和指数p稳定性问题。通过构造恰当的压缩算子并利用不动点定理,巧妙的应用Cp不等式、Burkholder-Davids-Gundy不等式,Ho¨lder不等式得到了所关注方程的稳定性结果。
其次研究带有非线性扰动的随机时滞微分系统(包括单时滞微分系统和混合时滞微分系统)的有界输入有界输出(BIBO)在均方意义下的稳定性质。利用Razumikhin方法和比较原理相结合,合理设计出状态反馈控制器的形式,给出了具有非线性扰动的随机时滞微分系统的均方BIBO稳定性条件。利用Lyapunov泛函理论并结合Riccati方程,有效的对控制器进行分解,得到了具有非线性扰动的混合时滞随机微分系统的时滞依赖的均方BIBO稳定性条件。利用不等式技巧和Lyapunov稳定性理论相结合,通过对系数矩阵的分解,对方程进行变形,建立合适的Lyapunov-Krasovskii泛函,以线性矩阵不等式(LMI)的形式给出了具有非线性扰动的时滞随机微分系统的时滞依赖的均方BIBO稳定性条件。
最后研究具有时滞(混合时滞和离散随机时滞及分布时滞)的离散时间随机神经网络(DSNNs)的稳定性。对具有混合时滞的DSNNs,我们根据时滞分离方法和自由权矩阵相结合,运用Schur补引理,得到保守性较低的时滞依赖的均方渐近稳定性条件。在DSNNs模型中,时滞很多时候是以随机的形式出现的,本文假定时滞为Bernoulli随机变量并考虑其概率分布,将其引入到DSNNs模型的参数矩阵中,通过建立扩充的Lyapunov-Krasovskii泛函,运用不等式分析技巧和自由权矩阵相结合,研究具有离散随机时滞和参数不确定的DSNNs,给出了时滞依赖的鲁棒均方指数稳定性条件。随后研究了具有混合时滞(离散随机时滞和分布时滞)的DSNNs,给出了时滞概率分布依赖的均方渐近稳定性条件。
对全文进行总结,并指出今后的研究方向。
As an important mathematic model, stochastic delay differential equations is deter-mine delay differential equations with random elements or stochastic differential equa-tions with time delays. That is, stochastic delay differential equations can model realproblems specifically. Therefore, they have been widely applied in many fields of sci-ence, such as automatic control, neural networks, biology, economics, chemical reactionengineering etc. Due to the instability comes from stochastic disturbance and delay ef-fects, It is very necessary to study the stability of stochastic delay differential systems.
The research backgrounds and research status on stability of stochastic delay differ-ential equations are introduced. It also provides the readers with some basic definitionsand important lemmas that are frequently used in this dissertation.
Firstly, the exponential p-stability for linear neutral stochastic differential equationswith variable delays, the exponential p-stability and asymptotically p-stability for linearneutral stochastic differential equations with mixed delays are studied respectively. Byconstructing suitable contraction mapping and employing fixed point theory, applying Cpinequality, Burkholder-Davids-Gundy inequality, Holder inequality, some novel stabilitycriteria are derived.
Secondly, the bounded-input bounded-output(BIBO) stability in mean square forstochastic differential equations with nonlinear perturbation and delays (discrete time de-lays and mixed time delays) is investigated. Using Razumikhin technique and comparisonprinciple to obtain the novel BIBO stabilization criteria in mean square for stochastic de-lay differential systems with nonlinear perturbation, based on this, the design of statefeedback is given by matrix transform. Combine Lyapunov function theory with Ric-cati equations to analyze controller, some delay-dependent stability criteria for stochasticdifferential equations with nonlinear perturbation and mixed delays are obtained. Makegood use of inequality and Lyapunov function theory, some novel delay-dependent BIBOstability criteria in mean square are derived and formulated in the form of linear matrixinequalities (LMIs) by constructing a new class of Lyapunov-Krasovskii functionals andthe descriptor model of the system and the method of decomposition.
Lastly, the stability of discrete-time stochastic neural networks(DSNNs) with de- lays(mixed time delays and discrete random time delays and distributed time delays)is studied. For DSNNs with mixed time delays, based on delay partitioning idea andfree-weighting matrix approach, applying Schur complement, a less conservative delay-dependent asymptotically stable criterion in the mean square can be developed. InDSNNs, the time delay is assumed to be Bernoulli stochastic vary and always ap-pears in random way. Then the information of the probability distribution of the time-varying delay is considered and transformed into parameter matrices of the transferredDSNN model, by constructing a new augmented Lyapunov-Krasovskii functional andintroducing some new analysis techniques, some delay-probability-distribution- depen-dent robust exponential stability criteria in mean square for DSNNS with randomlytime-varying delays and parameter uncertainties are derived, some delay-probability-distribution-dependent asymptotically stability criteria in mean square for DSNNS withdiscrete randomly time delays and distributed time delays are derived.
We summarize the main results obtained in this dissertation, and point out the futureworks that have been the author’s concerns.
简单介绍随机时滞微分方程稳定性研究背景和研究现状,给出了本文中常用的基本概念、基础知识和重要引理。
首先研究具有变时滞线性中立型随机微分方程的指数p稳定性以及具有混合变时滞线性中立型随机微分方程的渐近p稳定和指数p稳定性问题。通过构造恰当的压缩算子并利用不动点定理,巧妙的应用Cp不等式、Burkholder-Davids-Gundy不等式,Ho¨lder不等式得到了所关注方程的稳定性结果。
其次研究带有非线性扰动的随机时滞微分系统(包括单时滞微分系统和混合时滞微分系统)的有界输入有界输出(BIBO)在均方意义下的稳定性质。利用Razumikhin方法和比较原理相结合,合理设计出状态反馈控制器的形式,给出了具有非线性扰动的随机时滞微分系统的均方BIBO稳定性条件。利用Lyapunov泛函理论并结合Riccati方程,有效的对控制器进行分解,得到了具有非线性扰动的混合时滞随机微分系统的时滞依赖的均方BIBO稳定性条件。利用不等式技巧和Lyapunov稳定性理论相结合,通过对系数矩阵的分解,对方程进行变形,建立合适的Lyapunov-Krasovskii泛函,以线性矩阵不等式(LMI)的形式给出了具有非线性扰动的时滞随机微分系统的时滞依赖的均方BIBO稳定性条件。
最后研究具有时滞(混合时滞和离散随机时滞及分布时滞)的离散时间随机神经网络(DSNNs)的稳定性。对具有混合时滞的DSNNs,我们根据时滞分离方法和自由权矩阵相结合,运用Schur补引理,得到保守性较低的时滞依赖的均方渐近稳定性条件。在DSNNs模型中,时滞很多时候是以随机的形式出现的,本文假定时滞为Bernoulli随机变量并考虑其概率分布,将其引入到DSNNs模型的参数矩阵中,通过建立扩充的Lyapunov-Krasovskii泛函,运用不等式分析技巧和自由权矩阵相结合,研究具有离散随机时滞和参数不确定的DSNNs,给出了时滞依赖的鲁棒均方指数稳定性条件。随后研究了具有混合时滞(离散随机时滞和分布时滞)的DSNNs,给出了时滞概率分布依赖的均方渐近稳定性条件。
对全文进行总结,并指出今后的研究方向。
As an important mathematic model, stochastic delay differential equations is deter-mine delay differential equations with random elements or stochastic differential equa-tions with time delays. That is, stochastic delay differential equations can model realproblems specifically. Therefore, they have been widely applied in many fields of sci-ence, such as automatic control, neural networks, biology, economics, chemical reactionengineering etc. Due to the instability comes from stochastic disturbance and delay ef-fects, It is very necessary to study the stability of stochastic delay differential systems.
The research backgrounds and research status on stability of stochastic delay differ-ential equations are introduced. It also provides the readers with some basic definitionsand important lemmas that are frequently used in this dissertation.
Firstly, the exponential p-stability for linear neutral stochastic differential equationswith variable delays, the exponential p-stability and asymptotically p-stability for linearneutral stochastic differential equations with mixed delays are studied respectively. Byconstructing suitable contraction mapping and employing fixed point theory, applying Cpinequality, Burkholder-Davids-Gundy inequality, Holder inequality, some novel stabilitycriteria are derived.
Secondly, the bounded-input bounded-output(BIBO) stability in mean square forstochastic differential equations with nonlinear perturbation and delays (discrete time de-lays and mixed time delays) is investigated. Using Razumikhin technique and comparisonprinciple to obtain the novel BIBO stabilization criteria in mean square for stochastic de-lay differential systems with nonlinear perturbation, based on this, the design of statefeedback is given by matrix transform. Combine Lyapunov function theory with Ric-cati equations to analyze controller, some delay-dependent stability criteria for stochasticdifferential equations with nonlinear perturbation and mixed delays are obtained. Makegood use of inequality and Lyapunov function theory, some novel delay-dependent BIBOstability criteria in mean square are derived and formulated in the form of linear matrixinequalities (LMIs) by constructing a new class of Lyapunov-Krasovskii functionals andthe descriptor model of the system and the method of decomposition.
Lastly, the stability of discrete-time stochastic neural networks(DSNNs) with de- lays(mixed time delays and discrete random time delays and distributed time delays)is studied. For DSNNs with mixed time delays, based on delay partitioning idea andfree-weighting matrix approach, applying Schur complement, a less conservative delay-dependent asymptotically stable criterion in the mean square can be developed. InDSNNs, the time delay is assumed to be Bernoulli stochastic vary and always ap-pears in random way. Then the information of the probability distribution of the time-varying delay is considered and transformed into parameter matrices of the transferredDSNN model, by constructing a new augmented Lyapunov-Krasovskii functional andintroducing some new analysis techniques, some delay-probability-distribution- depen-dent robust exponential stability criteria in mean square for DSNNS with randomlytime-varying delays and parameter uncertainties are derived, some delay-probability-distribution-dependent asymptotically stability criteria in mean square for DSNNS withdiscrete randomly time delays and distributed time delays are derived.
We summarize the main results obtained in this dissertation, and point out the futureworks that have been the author’s concerns.
引文
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