Stability Of Impulsive Fractional-order Systems

Fractional-order calculus and integer-order calculus almost have the same long history, but research on the applications of fractional calculus can only be traced back to nearly twenty or thirty years from now.In recent years, the qualitative behavior of fractional-order dynamical system has become a hotspot of both applied areas and theoretical research. And because of the potential applications of analysis in models with hereditary and memory effects, the impulsive fractional-order system has become one of the frontier areas of applied mathematics. In general speaking, in integer-order system, the impulsive effects have big influences on the stability of the systems. So the stability of impulsive integer-order systems, including the stability of integer-order hybrid systems have long been concerned by researchers. As the author knows, the discussions of the stability of impulsive fractional-order systems are rare to see.This presentation mainly studies the stability of impulsive fractional order system. The content of it mainly includes the following four chapters:The first chapter gives the concept of fractional calculus and stability of impulsive fractional-order systems. In the first section, we mainly focus on the connections and differences between the three classical definitions of fractional order derivative, the memory characteristics of fractional order derivative, as well as the relationship with the multi-fractal; the second section introduces the concept of stability of fractional-order systems and impulsive fractional-order systems, also briefly introduces the current situation of research at home and abroad.The second chapter consists of some necessary preliminaries, including relevant background knowledge and some definitions and lemmas that are used in the later chapters.The third chapter studies the Mittag-Leffler stability and Ulam-Hyers stability for impulsive fractional-order systems. In the discussion of Mittag-Leffler stability, the theory of fractional-order systems is extended to the situation of impulsive fractional-order systems; in the discussion of Ulam-Hyers stability, the concept of Ulam-Hyers stability of functional differential equations is extended to the situation of impulsive fractional-order systems.The fourth chapter discusses two types of finite-time stability of impulsive fractional-order systems. The first class of finite-time stability can be used in the situation of predicting the stable-time of given systems, giving specified error range of initial value and allowed system error range. The second class of finite-time stability can be used to predict when the system error is reduced to zero, giving specified error range of initial value.