Analytic Study On The Stability For Several Kinds Of Fractional Differential Systems

In recent forty years,fractional-order calculus has played a more and more important role in many fields,such as dynamics?calorifics?biology?control and so on.Especially in recent years,a rapidly growing researchers pay much attention on the stability of fractional neural networks,such as asymptotical stability?finite-time stability and have made huge progress.How the delays influence the stability of the system is the key point when we investigate in the fractional neural networks,and there is little research attention on the stability for neural networks with delays and distributed delays.In this paper,the exponential estimate for the solution and the general solution of the fractional neutral differential equation with time delay are discussed,what's more,the global asymptotical stability of fractional neural networks with distributed delays and the finite-time stability of impulsive fractional-order neural networks with distributed delay are investigated.The main content of this paper is arranged as follows:In the first chapter,we first introduce the background and the significance of fractional-order calculus,then the pre-knowledge which is useful in the paper are presented.In the second chapter,the general solution and estimation for fractional neutral differential equations with time delay are investigated.Firstly,using step by step method,the existence and uniqueness of the solution for fractional neutral differential equation with time delay is discussed.Then,in the condition of the existence of the solution,the exponential estimation is given by Gronwall inequality.With the help of fundamental matrix and Laplace transformation,we get the general solution of the system.In the third chapter,the finite-time stability of impulsive fractional-order neural networks with distributed delay is investigated.Two sufficient conditions for the finite-time stability of impulsive fractional-order neural networks with distributed delay are given by Gronwall inequality.Then we present two simple examples to check the usefulness of the results.In the fourth chapter,the global asymptotical stability of Riemann-Liouville frac-tional neural networks with distributed delay is investigated.Two sufficient conditions for the global asymptotical stability of fractional neural networks with distributed delay are given by the method of Lyapunov function.Then we present two simple examples to check the usefulness of the results.