LMI方法在随机延迟微分方程中的应用
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摘要
如果一个系统某个时候的状态,受到众多因素的影响,而每个因素对系统的影响又具有很大的偶然性,则必须在系统中考虑环境噪声的影响.当考虑的系统其发展趋势不仅与现在的状态有关,还与过去的历史有关,则通常利用随机延迟微分方程来描述.随着科学研究的逐步深入,要求对实际系统的描述愈来愈精确.具有参数不确定的随机延迟微分方程、具有Markov调制的随机延迟微分方程、奇异随机延迟微分方程等为满足这一需要提供了新的数学工具,并且在控制理论、人工智能、网络分析、生态系统、化学、金融学等方面有许多重要应用.在随机微分方程理论中,解的估计和稳定性是两个基本问题.因此,本文对具有参数不确定性、具有Markov调制或奇异型的随机延迟微分方程解的估计和稳定性问题进行了分析和研究.
线性矩阵不等式(LMI)方法是研究系统稳定性和渐近行为的重要工具,因它是用代数量度来表示系统解的渐近或稳定性条件,因而更具可操作性和实用性.在实际工程应用中,LMI方法被广泛采用,特别针对不确定系统的分析和综合.本文主要研究LMI方法如何应用于处理随机延迟微分方程解的估计和稳定性问题.
本文综合运用Lyapunov-Krasovskii泛函、It?o公式、广义It?o公式及Gronwall不等式、Schur补不等式等多种技巧,考察了一类参数不确定的随机延迟微分方程,将其估计条件表示为一个LMI,从而得到了由上LMI可以确定的解的估计式.同时,基于LMI方法讨论了具有不确定参数的中立型随机延迟微分方程解的估计问题.另外,通过改进新的Lyapunov-Krasovskii泛函,首次利用LMI方法考察了同时具有Markov调制和参数不确定性的随机延迟微分方程解的估计,其估计表达式由LMI的解确定.为了阐述LMI方法在解决此类随机微分方程解的估计问题的有效性,分别给出了相应结果的数值应用.本文结果能涵盖现有参考文献中不存在随机扰动项的不确定微分方程的相关结果.
针对具有Markov调制的非线性随机延迟微分方程的稳定性进行了研究,由Lyapunov-Krasovskii泛函微分方程稳定性理论和现代概率论的相关理论,给出了这类随机混合系统与时滞大小无关的稳定性判别准则,其充分条件由LMI表出.同时,利用等价增广系统的方法,即将所研究的系统转换为一个等价的奇异系统,得到了时滞依赖的稳定性新判据.这两类稳定性判据各有所长,通过实例说明了其应用性.我们的结果涵盖并推广了现有参考文献的部分结果.
进一步地,研究了具有Markov调制的奇异型随机延迟微分方程,运用类似于分析非奇异型随机延迟微分方程的方法,建立了时滞独立和时滞依赖的稳定性判别准则.就作者所知,关于奇异微分方程时滞依赖稳定性的已有结果主要是针对确定性微分方程和线性随机微分方方程的,而针对非线性随机微分方程的情形尚属首次.
同时,LMI方法也经常被应用于确定性神经网络的稳定性研究.由于在实际神经系统中,神经信号信号传输是一个受随机因素影响的充满噪声的过程,同时时滞的存在对神经网络的性能也产生很大的影响,要对神经网络系统恰当地描述,从而进一步地设计、分析和应用,就需要考虑噪声和时滞的影响.本文应用LMI方法,给出了具有多个时滞的随机神经网络稳定的充分条件.这些条件是以LMI的形式表示的,在实际中便于验证和计算.此外,文中的结论是在不需要假定激活函数的可微型与单调性的条件下得到的,因而文中的模型是一类更广义的随机模型.本文所得的结论扩展了最近相关文献中神经网络稳定性的部分结果.
If the state of a system is caused by numerus factors which influence on the system in a stochastic way, then environment noise should be taken into account in the description of the system. In addition, when the development trend of this system is related with not only present state but also history or future state, the rule of this kind of systems is often described by stochastic delay differential equation. With the deep development of the science study, the description on practical systems is required to be better and better. To satisfy the requirement, some more special stochastic differential equations, such as stochastic delay differential equations with parameter uncertainty or Markovian Switching, singular stochastic delay differential equations etc. provide a new mathematical tool, which have widely applied in the domain of control theory, artificial intelligence, network theory, biology theory, finance etc. Among the stochastic differential equations theory, two basic problems are the estimations and stability of the solutions. Therefore, we shall focus on the study of the two problems on nonsingular and singular stochastic delay differential equations with parameter uncertainty or Markovian switching in this thesis.
An LMI approach is an important technique to study the asymptotic behavior and stability of the solutions of a system, because the use of algebraic measures in the analysis has the benefits of being simple computationally and more applicable. In actual engineering application, the LMI approach is widely used, especially in the analysis and synthesis for uncertain systems. Therefore, we focus on how the LMI approach is applied to deals with the problems for the estimations and stability of the solutions for stochastic delay differential equations.
By means of Lyapunov-Krasovskii stability theory on functional differential equations,Ito equation and generalized Ito equation, as well as Gronwall inequality and Schur complement inequality, the LMI-based estimate conditions are obtained for a class of stochastic delay differential equation with parameter uncertainty in this thesis, and hence estimates of the solutions are determined by the LMIs above. Also, we discuss the estimates of the solutions for neutral stochastic delay differential equations with parameter uncertainty basing on LMI approach. Meanwhile, the estimation of the solution of stochastic delay differential equations with Markov switching and parameter uncertainty is firstly investigated by constructing new Lyapunov-Krasovskii functional, and the estimates can be achieved by the solutions of a group of LMIs. Finally, some numerical examples is given to illustrate the efficiency of LMI approach on the problem of the estimates for this class of stochastic delay differential equations. And our conclusions include and extend the others' results in existing references for uncertain differential equations without stochastic term.
The stability of a class of nonlinear stochastic delay differential equation with Markovian switching is studied. Basing on Lyapunov-Krasovskii functional stability theory and modern probability theory, the thesis establishes delay-independent criterions on stability of the hybrid systems, where the sufficient conditions for stability is presented in terms with LMIs. Meanwhile, by constructing an equivalent augmented system, that is, a singular form representation of the system, new delay-dependent stability criterion is developed. And each of the two criterions has its strong points. Finally, some examples are provided to illustrate the applicability of the presented conclusions. Our results generalize some of existing results in the references.
Further on, the stability problem of singular stochastic delay differential equations with Markovian switching is investigated. And the delay-independent and delay-dependent stability criterions are established via the similar analysis for nonsingular stochastic delay differential equations. To the best knowledge of author, the known results of delay-dependent stability criterions are for deterministic nonsingular differential equations or linear stochastic singular differential equations , there are no such results for nonlinear stochastic singular differential equations.
In addition, LMI approach has often been adopted to study the stability of deterministic neural network. And in practical neural systems, neural signal transmission is a process filled with noise and always influenced by stochastic factors, also time delay greatly influences on performance of neural network. To describe neural network properly and then design, analysis and apply it, it is required to introduce noise and time delay into neural network. In this thesis, we obtain the sufficient conditions for the stability of multi-delayed stochastic neural network by using LMI approach. The conditions take the form os LMIs, so they are verifiable and computable efficiently. In addition, our results are established without restricting differentiability and monotonicity of the activation functions, hence our model is a more general model. The main results in this thesis are the generalization or extension of some recent results in the literature.
线性矩阵不等式(LMI)方法是研究系统稳定性和渐近行为的重要工具,因它是用代数量度来表示系统解的渐近或稳定性条件,因而更具可操作性和实用性.在实际工程应用中,LMI方法被广泛采用,特别针对不确定系统的分析和综合.本文主要研究LMI方法如何应用于处理随机延迟微分方程解的估计和稳定性问题.
本文综合运用Lyapunov-Krasovskii泛函、It?o公式、广义It?o公式及Gronwall不等式、Schur补不等式等多种技巧,考察了一类参数不确定的随机延迟微分方程,将其估计条件表示为一个LMI,从而得到了由上LMI可以确定的解的估计式.同时,基于LMI方法讨论了具有不确定参数的中立型随机延迟微分方程解的估计问题.另外,通过改进新的Lyapunov-Krasovskii泛函,首次利用LMI方法考察了同时具有Markov调制和参数不确定性的随机延迟微分方程解的估计,其估计表达式由LMI的解确定.为了阐述LMI方法在解决此类随机微分方程解的估计问题的有效性,分别给出了相应结果的数值应用.本文结果能涵盖现有参考文献中不存在随机扰动项的不确定微分方程的相关结果.
针对具有Markov调制的非线性随机延迟微分方程的稳定性进行了研究,由Lyapunov-Krasovskii泛函微分方程稳定性理论和现代概率论的相关理论,给出了这类随机混合系统与时滞大小无关的稳定性判别准则,其充分条件由LMI表出.同时,利用等价增广系统的方法,即将所研究的系统转换为一个等价的奇异系统,得到了时滞依赖的稳定性新判据.这两类稳定性判据各有所长,通过实例说明了其应用性.我们的结果涵盖并推广了现有参考文献的部分结果.
进一步地,研究了具有Markov调制的奇异型随机延迟微分方程,运用类似于分析非奇异型随机延迟微分方程的方法,建立了时滞独立和时滞依赖的稳定性判别准则.就作者所知,关于奇异微分方程时滞依赖稳定性的已有结果主要是针对确定性微分方程和线性随机微分方方程的,而针对非线性随机微分方程的情形尚属首次.
同时,LMI方法也经常被应用于确定性神经网络的稳定性研究.由于在实际神经系统中,神经信号信号传输是一个受随机因素影响的充满噪声的过程,同时时滞的存在对神经网络的性能也产生很大的影响,要对神经网络系统恰当地描述,从而进一步地设计、分析和应用,就需要考虑噪声和时滞的影响.本文应用LMI方法,给出了具有多个时滞的随机神经网络稳定的充分条件.这些条件是以LMI的形式表示的,在实际中便于验证和计算.此外,文中的结论是在不需要假定激活函数的可微型与单调性的条件下得到的,因而文中的模型是一类更广义的随机模型.本文所得的结论扩展了最近相关文献中神经网络稳定性的部分结果.
If the state of a system is caused by numerus factors which influence on the system in a stochastic way, then environment noise should be taken into account in the description of the system. In addition, when the development trend of this system is related with not only present state but also history or future state, the rule of this kind of systems is often described by stochastic delay differential equation. With the deep development of the science study, the description on practical systems is required to be better and better. To satisfy the requirement, some more special stochastic differential equations, such as stochastic delay differential equations with parameter uncertainty or Markovian Switching, singular stochastic delay differential equations etc. provide a new mathematical tool, which have widely applied in the domain of control theory, artificial intelligence, network theory, biology theory, finance etc. Among the stochastic differential equations theory, two basic problems are the estimations and stability of the solutions. Therefore, we shall focus on the study of the two problems on nonsingular and singular stochastic delay differential equations with parameter uncertainty or Markovian switching in this thesis.
An LMI approach is an important technique to study the asymptotic behavior and stability of the solutions of a system, because the use of algebraic measures in the analysis has the benefits of being simple computationally and more applicable. In actual engineering application, the LMI approach is widely used, especially in the analysis and synthesis for uncertain systems. Therefore, we focus on how the LMI approach is applied to deals with the problems for the estimations and stability of the solutions for stochastic delay differential equations.
By means of Lyapunov-Krasovskii stability theory on functional differential equations,Ito equation and generalized Ito equation, as well as Gronwall inequality and Schur complement inequality, the LMI-based estimate conditions are obtained for a class of stochastic delay differential equation with parameter uncertainty in this thesis, and hence estimates of the solutions are determined by the LMIs above. Also, we discuss the estimates of the solutions for neutral stochastic delay differential equations with parameter uncertainty basing on LMI approach. Meanwhile, the estimation of the solution of stochastic delay differential equations with Markov switching and parameter uncertainty is firstly investigated by constructing new Lyapunov-Krasovskii functional, and the estimates can be achieved by the solutions of a group of LMIs. Finally, some numerical examples is given to illustrate the efficiency of LMI approach on the problem of the estimates for this class of stochastic delay differential equations. And our conclusions include and extend the others' results in existing references for uncertain differential equations without stochastic term.
The stability of a class of nonlinear stochastic delay differential equation with Markovian switching is studied. Basing on Lyapunov-Krasovskii functional stability theory and modern probability theory, the thesis establishes delay-independent criterions on stability of the hybrid systems, where the sufficient conditions for stability is presented in terms with LMIs. Meanwhile, by constructing an equivalent augmented system, that is, a singular form representation of the system, new delay-dependent stability criterion is developed. And each of the two criterions has its strong points. Finally, some examples are provided to illustrate the applicability of the presented conclusions. Our results generalize some of existing results in the references.
Further on, the stability problem of singular stochastic delay differential equations with Markovian switching is investigated. And the delay-independent and delay-dependent stability criterions are established via the similar analysis for nonsingular stochastic delay differential equations. To the best knowledge of author, the known results of delay-dependent stability criterions are for deterministic nonsingular differential equations or linear stochastic singular differential equations , there are no such results for nonlinear stochastic singular differential equations.
In addition, LMI approach has often been adopted to study the stability of deterministic neural network. And in practical neural systems, neural signal transmission is a process filled with noise and always influenced by stochastic factors, also time delay greatly influences on performance of neural network. To describe neural network properly and then design, analysis and apply it, it is required to introduce noise and time delay into neural network. In this thesis, we obtain the sufficient conditions for the stability of multi-delayed stochastic neural network by using LMI approach. The conditions take the form os LMIs, so they are verifiable and computable efficiently. In addition, our results are established without restricting differentiability and monotonicity of the activation functions, hence our model is a more general model. The main results in this thesis are the generalization or extension of some recent results in the literature.
引文
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