Research On Stability Of Impulsive Neural Networks Under Inverse Lipschitz Condition

In recent years, the dynamics of impulsive time delay neural networks is the study hot of neural networks. In these studies, we often assume that some parts of the neural network, such as the activation function and behavior functions, meet the conditions of continuity, global or local Lipschitz. This assumption is sometimes harsh. For instance, modeling some facts in studing neural computation, we need other condition, such as inverse Lipschitz, to meet. Under the inverse Lipschitz condition, this paper studied the exponential stability of impulsive time delay neural networks, and achieved good results. The main contents are as follows①The stability mathematical foundation of impulsive time delay neural networksImpulsive time delay differential equation is the theoretical basis of impulsive time delay neural networks. For the completely, the paper describes the related theorems of differential equations, such as its existence, uniqueness and stability. We also introduce a few of important differential inequalities and Lyapunov stability theory.②The research advance of impulsive time delay neural networksWe introduce the basic concept of impulse and time delay neural networks, the corresponding differential model, and the stability theorems commonly used to determine. Finally, we give a brief review of the development state for each type.③The stability of inverse Lipschitz variable-time impulse Cohen-Grossberg neural networkWe analyze the time delay neural networks, in which the continous part is divergent, but as a whole it is exponential convergent by mixing impulses. Impulses occur at variable time which depends on states. The paper also discusses the"beating phenomenon"of impulses. We separately propose the sufficient conditions of the exponential stability by whether it has time delay. According to two different conditions, we give its corresponding stability theory for the latter. We also discuss the system, in which behavior functions are in inverse Lipschitz conditions. Four numerical examples demonstrate the effectiveness of the results, which have no time-delay, second-order and third-order time-delay system, respectively.④The stability of inverse Lipschitz continuous time impulse Cohen-Grossberg neural network The Cohen-Grossberg neural network, whose activation function is in the inverse Lipschitz condition, is analyzed. In this case, the existence and uniqueness of equations solution is proved by the topological degree theory. To simplify the computing of topological degree, we construct a continuous homotopic map and employ expanded Jocobian matrix set and its theory. The paper proposes some global exponential stability sufficient conditions. Two numerical examples are given to verity the effectiveness of the results.⑤The stability of inverse Lipschitz discrete time impulse Cohen-Grossberg neural networkThe paper analyse the global exponential stability and global asymptotic stability, in which behavior functions are in inverse Lipschitz conditions and their time is discrete. We discuss separately the stability by whether it has time delay. When the time delay is regular, we simplify the condition. We prove it mainly using discrete Halanay inequality. Three numerical examples demonstrate the effectiveness of the results, which have no time-delay, second-order and third-order time-delay system, respectively. And they also demonstrate that those theorems are effective even though the impulse has enlarged effect, provided not exceeding a certain limit.