Stability Of Several Classes Of Fractional Impulsive Differential Equations

Fractional impulsive differential equation theory has a wide range of applications in the fields of physics,chemical engineering,fluid mechanics,cybernetics and so on,and has received extensive attention in recent years.The stability theory of fractional impulsive differential equations is the basis of studying various fractional impulsive dynamical systems.In this dissertation,we explore the stability of several kinds of fractional impulsive differential equations.The main research contents and innovations are as follows:1.For a new class of impulsive fractional non-autonomous network coupled systems based on directed graph theory,a global Lyapunov function is constructed using directed graph theory,and two new Mittag-Leffler inequalities are proved,which resolve the problem that Laplace transform is inapplicable to fractional impulsive differential equations.Furthermore,the stability principle of this type of equations is obtained by applying the basic theory of strongly connected directed graph,new inequalities and Lyapunov method.These principles are applied to fractional impulsive coupled non-autonomous systems and stability criteria including stability,uniform stability,and Mittag-Leffler stability are obtained.2.For the globally S-asymptotic ω-periodicity of a class of impulsive non-autonomous fractional-order neural networks,based on the definition of the new piecewise S-asymptoticω-periodic function,it is proved that the function class is Banach space.Based on the Banach contracting mapping principle,the existence and uniqueness of the piecewise Sasymptotic ω-periodic solutions of this class of fractional-order impulsive differential equations are obtained.The globally asymptotic stability of S-asymptotic ω-periodic solutions of this type of equations is given by using fractional differential and integral inequalities.3.For a class of fractional complex-valued neural network system with linear impulses and fixed delays,the constraint that the first-order parameter is less than 1 is overcome.By using symbolic function and Banach fixed point theorem,the uniform stability criterion and the existence and uniqueness criterion of equilibrium solution for this system are obtained.In particular,when a solution of the neural network is uniformly stable,the solution is the only equilibrium solution.4.For a class of fractional nonlinear differential systems with state-dependent delayed impulses,a discriminant lemma for the monotonicity for solutions of fractional differential equations is given.Based on this lemma,linear matrix inequalities and several comparative arguments,judgment conditions of globally/locally uniform stability,globally/locally uniform asymptotic stability and globally/locally Mittag-Leffler stability of the solutions of this class of systems are obtained.In this dissertation,the stability of several kinds of fractional impulsive differential equations is studied,which extends the results of integral order correlated stability,enriches and develops the theoretical content of fractional impulsive differential equations,and provides the necessary theoretical basis for their application in practical problems.