Dynamics Analysis Of Impulsive Functional Differential Systems And Their Application In Neural Networks

In recent years. impulsive functional differential systems have been widely used in neural networks, optical control, population dynamics, biotechnology, economics and other fields, It has become a hot research topic and also attracts many mathematician's attention. and various results for the impulsive functional differential system are obtained . However, the research in those fields is not enough, for instance, does the invariance principle can extend to non-autonomous system under impulsive disturbance? Are there some more general results on oscillation of impulsive functional differential system of mixed type with advanced constant argument? Are there some sufficient conditions that guarantee the global stability of impulsive functional differential system with infinite delays? and so on. Hence, there are a lot of work we need to do in this field. In this paper. we focus on the research work on dynamics analysis of impulsive functional differential systems and give some affirmative answers for above problems. This paper is divided into three parts.In chapter one. we study the uniform asymptotic stability, global exponential stability and W-stability of impulsive functional differential systems with finite delays as follows:We investigate the weak exponential stability and global exponential stability of system (1) by establishing the extended Halanay' s 1-dimension delay differential inequality. Besides, we study the stability of system (1) by using Lyapunov functions and Razumikhin technique. Some new Razumikhin type theorems on uniform stability and uniform asymptotic stability are obtained here. Also, some application on neural networks is given to illustrate the advantages of the obtained results on chapter three. On the other hand, we discuss the invariance principle that can extend to non-autonomous system under the effects of impulses and delays. We investigate the W- stability and W-uniform stability of impulsive functional differential systems and preliminary establish the relation between uniform stability, asymptotic stability and W-stability. In chapter two. we study the oscillation and asymptotic property of impulsive delay differential equation with piecewise constant argument of advanced type as follows:where Z_+ is the set of all positive integers, [·] denotes set of maximum integers,σis any quotient of positive odd integers, m > 0, m∈Z_+. By establishing auxiliary function and using analysis technique, we analyze and investigate the oscillation and asymptotic property of system (2) with non-positive coefficient or non-negative coefficient. There are two sections: when a_i,e,p are non-negative functions, we obtain some results ensuring the bounded solution to be oscillatory; when a_i, e, p are non-positive functions, we obtain some results ensuring the bounded solution either to oscillate or to tend to zero. Besides, we analyze and investigate above results with the change ofσ. Our results in this chapter generalize and improve several known results.In chapter three, we obtain a criterion for the uniform asymptotic stability of the equilibrium point of impulsive delayed Hopfield neural networks by using Lyapunov functions and some obtained results in chapter one. Besides, some new sufficient conditions ensuring exponential stability of the equilibrium point of impulsive delayed Hopfield neural networks are obtained by using linear matrix inequality approach (LMI). Our results in this chapter generalize and improve several known results. which is beneficial to practical application.