随机时滞反应扩散Hopfield神经网络的渐近性分析
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摘要
本文研究了一系列随机时滞反应扩散Hopfield神经网络的渐近行为,主要分成以下六部分.
第一章,不仅回顾了神经网络研究的历史和现状，而且论述了研究时滞随机反应扩散Hopfield神经网络的理论意义和应用价值.同时,指出引入时滞，扩散，和随机扰动后系统出现的新现象，并阐述了我们将会遇到的数学困难.
第二章,研究由无穷维Wiener过程驱动的时滞反应扩散Hopfield神经网络的长期动力行为.首先研究解的存在唯一性.接着给出系统随机指数稳定的条件.最后给出例题验证结果的有效性,同时用Matlab给出仿真.
第三章,研究由Brown运动驱动的时滞反应扩散Hopfield神经网络的渐近行为.利用Poincare′不等式和随机分析技巧,给出了系统均方意义下指数稳定性的充分条件.同时利用Burkholder-Davis-Gundy不等式, Chebyshev不等式以及Borel-Cantelli引理给出了系统几乎必然指数稳定性的判据.最后给出例题验证结果和方法的有效性,同样用Matlab给出仿真.
第四章，首先研究了随机时滞反应扩散Hopfield神经网络(RDHNN)的温和解的全局存在唯一性,紧接着通过构造合适的随机动力系统证明了随机吸引子的存在性.
第五章研究了一类具有S-分布时滞和反应扩散项的Hopfield神经网络的滑动模控制问题.首先改进了一类Hanalay不等式,给出了一种范数不等式.然后通过等效控制方法建立了系统的滑动模态方程,并利用不等式技巧分析了它的吸引集的存在性和零点的指数稳定性.在此基础上设计了变结构控制器,给出了运动轨线到达滑动模态区的时间估计.最后给出了一个例子验证了本文的结果,并利用Matlab作出了仿真.
最后，我们给出了研究前景展望.
In this paper, we study the asymptotic behavior for different kinds of stochasticreaction diffusion Hopfield neural network with delay, and this dissertation is dividedinto6parts.
In the first chapter, we not only review history and present situation of researchon the neural network, but also explain theoretical significance and practical valueto the research of neural network. Moreover, We will discuss the new phenomenonsaccompanying the occurrence of delay, reaction-diffusion term as well as stochas-tic disturbance, meanwhile the corresponding mathematical difficulties that we willmeet will be given in this chapter.
In the second chapter, we focus on the long time behavior of the mild solution fordelayed reaction-diffusion Hopfield neural networks driven by infinite dimensionalWiener processes. We will first study the existence and uniqueness of the mild so-lution for this system. Afterwards we will prove the stochastic exponential stabilityof this system. An example is given to examine the availability of the results of thispaper, simulations is also given by using the Matlab.
In the third chapter, we study the asymptotic behavior for a class of delayedreaction-diffusion Hopfield neural networks driven by finite dimensional Wiener pro-cesses. Some new sufficient conditions are established to guarantee the mean squareexponential stability of this system by using Poincare′’s inequality and stochasticanalysis technique. The proof of the almost surely exponential stability for this sys-tem is carried out by using the Burkholder-Davis-Gundy inequality, the Chebyshevinequality as well as the Borel-Cantelli lemma. Finally, an example is given to illus-trate the effectiveness of the proposed approach, and the simulation is also given byusing the Matlab.
In the fourth chapter, we study the global existence and uniqueness of the mildsolution for reaction-diffusion Hopfield neural networks (RDHNN) driven by Wiener processes using a priori estimate, then the random attractor for this system is alsoestablished by constructing proper random dynamical system.
In the fifth chapter, the sliding mode control problems of a class of Hopfield neu-ral networks with S-type distributed delays and reaction diffusion terms are investi-gated. First, the improved Hanalay inequality and norm inequality are presented.Then, the sliding mode equation is established by the equivalent control method,and the existence of the attracting sets and exponential stability of this system arediscussed by using the inequalities. Then the variable controller is designed, the ap-proximate time estimate from any initial state to sliding manifolds is also obtained.Finally, an example is given to prove the availability the result of this paper, and thesimulation is also given by using the Matlab.
The prospect of our research will be given at last.
第一章,不仅回顾了神经网络研究的历史和现状，而且论述了研究时滞随机反应扩散Hopfield神经网络的理论意义和应用价值.同时,指出引入时滞，扩散，和随机扰动后系统出现的新现象，并阐述了我们将会遇到的数学困难.
第二章,研究由无穷维Wiener过程驱动的时滞反应扩散Hopfield神经网络的长期动力行为.首先研究解的存在唯一性.接着给出系统随机指数稳定的条件.最后给出例题验证结果的有效性,同时用Matlab给出仿真.
第三章,研究由Brown运动驱动的时滞反应扩散Hopfield神经网络的渐近行为.利用Poincare′不等式和随机分析技巧,给出了系统均方意义下指数稳定性的充分条件.同时利用Burkholder-Davis-Gundy不等式, Chebyshev不等式以及Borel-Cantelli引理给出了系统几乎必然指数稳定性的判据.最后给出例题验证结果和方法的有效性,同样用Matlab给出仿真.
第四章，首先研究了随机时滞反应扩散Hopfield神经网络(RDHNN)的温和解的全局存在唯一性,紧接着通过构造合适的随机动力系统证明了随机吸引子的存在性.
第五章研究了一类具有S-分布时滞和反应扩散项的Hopfield神经网络的滑动模控制问题.首先改进了一类Hanalay不等式,给出了一种范数不等式.然后通过等效控制方法建立了系统的滑动模态方程,并利用不等式技巧分析了它的吸引集的存在性和零点的指数稳定性.在此基础上设计了变结构控制器,给出了运动轨线到达滑动模态区的时间估计.最后给出了一个例子验证了本文的结果,并利用Matlab作出了仿真.
最后，我们给出了研究前景展望.
In this paper, we study the asymptotic behavior for different kinds of stochasticreaction diffusion Hopfield neural network with delay, and this dissertation is dividedinto6parts.
In the first chapter, we not only review history and present situation of researchon the neural network, but also explain theoretical significance and practical valueto the research of neural network. Moreover, We will discuss the new phenomenonsaccompanying the occurrence of delay, reaction-diffusion term as well as stochas-tic disturbance, meanwhile the corresponding mathematical difficulties that we willmeet will be given in this chapter.
In the second chapter, we focus on the long time behavior of the mild solution fordelayed reaction-diffusion Hopfield neural networks driven by infinite dimensionalWiener processes. We will first study the existence and uniqueness of the mild so-lution for this system. Afterwards we will prove the stochastic exponential stabilityof this system. An example is given to examine the availability of the results of thispaper, simulations is also given by using the Matlab.
In the third chapter, we study the asymptotic behavior for a class of delayedreaction-diffusion Hopfield neural networks driven by finite dimensional Wiener pro-cesses. Some new sufficient conditions are established to guarantee the mean squareexponential stability of this system by using Poincare′’s inequality and stochasticanalysis technique. The proof of the almost surely exponential stability for this sys-tem is carried out by using the Burkholder-Davis-Gundy inequality, the Chebyshevinequality as well as the Borel-Cantelli lemma. Finally, an example is given to illus-trate the effectiveness of the proposed approach, and the simulation is also given byusing the Matlab.
In the fourth chapter, we study the global existence and uniqueness of the mildsolution for reaction-diffusion Hopfield neural networks (RDHNN) driven by Wiener processes using a priori estimate, then the random attractor for this system is alsoestablished by constructing proper random dynamical system.
In the fifth chapter, the sliding mode control problems of a class of Hopfield neu-ral networks with S-type distributed delays and reaction diffusion terms are investi-gated. First, the improved Hanalay inequality and norm inequality are presented.Then, the sliding mode equation is established by the equivalent control method,and the existence of the attracting sets and exponential stability of this system arediscussed by using the inequalities. Then the variable controller is designed, the ap-proximate time estimate from any initial state to sliding manifolds is also obtained.Finally, an example is given to prove the availability the result of this paper, and thesimulation is also given by using the Matlab.
The prospect of our research will be given at last.
引文
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