A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations

Abstract

Abstract We present an overview of the literature on solutions to impulsive Caputo fractional differential equations. Lyapunov direct method is used to obtain sufficient conditions for stability properties of the zero solution of nonlinear impulsive fractional differential equations. One of the main problems in the application of Lyapunov functions to fractional differential equations is an appropriate definition of its derivative among the differential equation of fractional order. A brief overview of those used in the literature is given, and we discuss their advantages and disadvantages. One type of derivative, the so called Caputo fractional Dini derivative, is generalized to impulsive fractional differential equations. We apply it to study stability and uniform stability. Some examples are given to illustrate the results.