本文選題：不連續(xù)激勵(lì)函數(shù) + 混合時(shí)滯　； 參考：《湖南大學(xué)》2015年博士論文

【摘要】：本文通過運(yùn)用拓?fù)涠壤碚?多值版本的Leray-Schauder選擇定理,不動(dòng)點(diǎn)定理,不等式技巧,Lyapunov泛函及矩陣?yán)碚摰认嘟Y(jié)合的方法對(duì)幾類具有混合時(shí)滯(即同時(shí)具有時(shí)變時(shí)滯和分布時(shí)滯)和不連續(xù)激勵(lì)函數(shù)的神經(jīng)網(wǎng)絡(luò)模型的動(dòng)力學(xué)性態(tài)進(jìn)行了研究,討論了這些網(wǎng)絡(luò)模型平衡點(diǎn)或概周期解的存在性,唯一性,全局穩(wěn)定性,輸出解的收斂性,有限時(shí)間一致收斂性等等.我們的結(jié)論不但削弱了眾多結(jié)果中對(duì)激勵(lì)函數(shù)的限制,而且推廣了已有文獻(xiàn)的相關(guān)結(jié)論,從而對(duì)神經(jīng)網(wǎng)絡(luò)的設(shè)計(jì)有重要的指導(dǎo)意義.本文做了如下幾個(gè)方面的工作:首先,我們利用多值版本的Leray-Schauder選擇定理,廣義李雅普諾夫泛函和不等式等方法研究了一類具有混合時(shí)滯(即同時(shí)具有時(shí)變時(shí)滯和分布時(shí)滯)和不連續(xù)激勵(lì)函數(shù)的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)模型,獲得了該系統(tǒng)的狀態(tài)變量的平衡點(diǎn)存在性,唯一性及全局指數(shù)穩(wěn)定的充分條件,而且討論了輸出解的收斂性.此處,激勵(lì)函數(shù)可以是無界的、非單調(diào)的,甚至激勵(lì)函數(shù)在其不連續(xù)點(diǎn)的左極限并不需要小于右極限,這在其他關(guān)于具有不連續(xù)激勵(lì)函數(shù)的Cohen-Grossberg神經(jīng)網(wǎng)絡(luò)的文獻(xiàn)中是少見的.所得結(jié)果不但推廣了具有滿足利普希茨條件的激勵(lì)函數(shù)的Cohen Grossberg神經(jīng)網(wǎng)絡(luò)的相關(guān)結(jié)果,而且對(duì)具有不連續(xù)激勵(lì)函數(shù)和常時(shí)滯的神經(jīng)網(wǎng)絡(luò)的相關(guān)結(jié)果也進(jìn)行了推廣.數(shù)值模擬的結(jié)果與我們的結(jié)論一致.其次,我們研究了一類推廣的具有混合時(shí)滯(即同具有時(shí)變時(shí)滯和分布時(shí)滯)和不連續(xù)激勵(lì)函數(shù)的競(jìng)爭(zhēng)神經(jīng)網(wǎng)絡(luò)模型.在放松已有文獻(xiàn)所要求的條件下,沒有假定激勵(lì)函數(shù)有界、單調(diào)及激勵(lì)函數(shù)在不連續(xù)點(diǎn)的左極限小于右極限,首先用多值版本的Leray-Schauder選擇定理、廣義李雅普諾夫泛函等方法獲得網(wǎng)絡(luò)模型的狀態(tài)變量的平衡點(diǎn)存在性,唯一性及全局穩(wěn)定的LMI型充分條件,研究了輸出解的收斂性;其次,利用M-矩陣的性質(zhì)、集值映射的拓?fù)涠壤碚摵蛷V義李雅普諾夫泛函等方法獲得網(wǎng)絡(luò)模型平衡點(diǎn)存在和全局指數(shù)穩(wěn)定的M型充分條件;最后,由于激勵(lì)函數(shù)的不連續(xù)性,本文研究了網(wǎng)絡(luò)模型的有限時(shí)間收斂性,而這一性質(zhì)的相關(guān)研究在競(jìng)爭(zhēng)神經(jīng)網(wǎng)絡(luò)模型中還不多見.另外,在激勵(lì)函數(shù)單調(diào)非減的條件下我們獲得平衡點(diǎn)的全局指數(shù)穩(wěn)定的充分條件.本章結(jié)果對(duì)已有文獻(xiàn)相關(guān)結(jié)論進(jìn)行了推廣和完善.數(shù)值模擬驗(yàn)證了所得結(jié)論.最后,在激勵(lì)函數(shù)單調(diào)非減、無界的前提下,我們利用矩陣?yán)碚�、不�?dòng)點(diǎn)理論和廣義Lyapunov泛函等方法首次研究了一類具有混合時(shí)滯(即同具有時(shí)變時(shí)滯和分布時(shí)滯)和不連續(xù)激勵(lì)函數(shù)的Cohen Grossberg神經(jīng)網(wǎng)絡(luò)模型的概周期解的動(dòng)力學(xué)性質(zhì),主要包括概周期解的存在性、全局穩(wěn)定性及全局指數(shù)漸近穩(wěn)定性等.所得結(jié)論是相關(guān)文獻(xiàn)關(guān)于周期解,平衡點(diǎn)相應(yīng)動(dòng)力學(xué)性質(zhì)的推廣.數(shù)值模擬與我們的結(jié)論相符.
[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.
【學(xué)位授予單位】：湖南大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2015
【分類號(hào)】：O175

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