几类具有不连续激励函数神经网络模型的动力学研究

发布时间：2018-06-01 23:44

本文选题：不连续激励函数 + 混合时滞　；参考：《湖南大学》2015年博士论文

【摘要】：本文通过运用拓扑度理论,多值版本的Leray-Schauder选择定理,不动点定理,不等式技巧,Lyapunov泛函及矩阵理论等相结合的方法对几类具有混合时滞(即同时具有时变时滞和分布时滞)和不连续激励函数的神经网络模型的动力学性态进行了研究,讨论了这些网络模型平衡点或概周期解的存在性,唯一性,全局稳定性,输出解的收敛性,有限时间一致收敛性等等.我们的结论不但削弱了众多结果中对激励函数的限制,而且推广了已有文献的相关结论,从而对神经网络的设计有重要的指导意义.本文做了如下几个方面的工作:首先,我们利用多值版本的Leray-Schauder选择定理,广义李雅普诺夫泛函和不等式等方法研究了一类具有混合时滞(即同时具有时变时滞和分布时滞)和不连续激励函数的Cohen-Grossberg神经网络模型,获得了该系统的状态变量的平衡点存在性,唯一性及全局指数稳定的充分条件,而且讨论了输出解的收敛性.此处,激励函数可以是无界的、非单调的,甚至激励函数在其不连续点的左极限并不需要小于右极限,这在其他关于具有不连续激励函数的Cohen-Grossberg神经网络的文献中是少见的.所得结果不但推广了具有满足利普希茨条件的激励函数的Cohen Grossberg神经网络的相关结果,而且对具有不连续激励函数和常时滞的神经网络的相关结果也进行了推广.数值模拟的结果与我们的结论一致.其次,我们研究了一类推广的具有混合时滞(即同具有时变时滞和分布时滞)和不连续激励函数的竞争神经网络模型.在放松已有文献所要求的条件下,没有假定激励函数有界、单调及激励函数在不连续点的左极限小于右极限,首先用多值版本的Leray-Schauder选择定理、广义李雅普诺夫泛函等方法获得网络模型的状态变量的平衡点存在性,唯一性及全局稳定的LMI型充分条件,研究了输出解的收敛性;其次,利用M-矩阵的性质、集值映射的拓扑度理论和广义李雅普诺夫泛函等方法获得网络模型平衡点存在和全局指数稳定的M型充分条件;最后,由于激励函数的不连续性,本文研究了网络模型的有限时间收敛性,而这一性质的相关研究在竞争神经网络模型中还不多见.另外,在激励函数单调非减的条件下我们获得平衡点的全局指数稳定的充分条件.本章结果对已有文献相关结论进行了推广和完善.数值模拟验证了所得结论.最后,在激励函数单调非减、无界的前提下,我们利用矩阵理论、不动点理论和广义Lyapunov泛函等方法首次研究了一类具有混合时滞(即同具有时变时滞和分布时滞)和不连续激励函数的Cohen Grossberg神经网络模型的概周期解的动力学性质,主要包括概周期解的存在性、全局稳定性及全局指数渐近稳定性等.所得结论是相关文献关于周期解,平衡点相应动力学性质的推广.数值模拟与我们的结论相符.
[Abstract]:In this paper, by using the topological degree theory, the multi-valued version of the Leray-Schauder selection theorem, the fixed point theorem, The dynamic behavior of several neural network models with mixed time-delay (i.e., time-varying delay and distributed time-delay) and discontinuous excitation function is studied by combining Lyapunov functional and matrix theory. In this paper, the existence, uniqueness, global stability, convergence of output solutions and finite-time uniform convergence of equilibrium or almost periodic solutions of these network models are discussed. Our conclusion not only weakens the limitation of excitation function in many results, but also generalizes the related conclusions in previous literatures, which is of great significance to the design of neural networks. In this paper, we do the following work: first, we use the multi-valued version of the Leray-Schauder selection theorem, Generalized Lyapunov Functionals and inequalities are used to study a class of Cohen-Grossberg neural network models with mixed delays (i.e., time-varying delays and distributed delays) and discontinuous excitation functions. Sufficient conditions for the existence, uniqueness and global exponential stability of the state variables of the system are obtained, and the convergence of the output solution is discussed. Here, the excitation function may be unbounded, nonmonotone, and even the left limit of the excitation function at its discontinuous point does not need to be less than the right limit, which is rare in other literatures on Cohen-Grossberg neural networks with discontinuous excitation functions. The results obtained not only generalize the related results of Cohen Grossberg neural networks with excitation function satisfying Lipschitz condition, but also extend the results of neural networks with discontinuous excitation function and constant delay. The numerical simulation results are in agreement with our conclusions. Secondly, we study a class of generalized competitive neural network models with mixed delay (that is, the same time-varying delay and distributed delay) and discontinuous excitation function. Under the condition of relaxing the existing literature, the bounded excitation function is not assumed. The left limit of monotone and the excitation function at discontinuous point is less than the right limit. Firstly, the multivalued version of Leray-Schauder selection theorem is used. Generalized Lyapunov Functionals and other methods are used to obtain the sufficient conditions for the existence, uniqueness and global stability of the equilibrium point of the state variables of the network model, and the convergence of the output solutions is studied. Secondly, the properties of the M- matrix are used. The topological degree theory of set-valued mappings and the generalized Lyapunov functional method are used to obtain M-type sufficient conditions for the existence of equilibrium points and global exponential stability of the network model. In this paper, the finite time convergence of the network model is studied, but the related research of this property is rare in the competitive neural network model. In addition, we obtain a sufficient condition for the global exponential stability of the equilibrium point under the condition that the excitation function is monotone and non-subtractive. The results of this chapter generalize and perfect the related conclusions of the literature. The results are verified by numerical simulation. Finally, on the premise that the excitation function is monotone, non-subtractive and unbounded, we use matrix theory. In this paper, the dynamical properties of almost periodic solutions of a class of Cohen Grossberg neural network models with mixed delay (that is, the same time-varying delay and distributed delay) and discontinuous excitation function are studied for the first time by using fixed point theory and generalized Lyapunov functional methods. It mainly includes the existence of almost periodic solutions, global stability and global exponential asymptotic stability. The conclusion is a generalization of the dynamical properties of periodic solutions and equilibrium points. The numerical simulation is in agreement with our conclusion.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O175

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