The Stability Of The Fractional Order Systems

Fractional order calculus is a new field in mathematic study, which is an expansionof traditional integral calculus theory. So far, a great of achievement has been attained inthe study of fractional order calculus theory, which provides a new theory foundationfor the application of fractional order calculus in other subjects. This paper considersthe problems of existence and uniqueness theorems and the finite-time stability analysisfor the initial value problem of linear fractional-order with multi-state time delaysystem.Firstly, we give the initial value problem which the paper studied and introducesome common basic functions, which is used to study fractional calculus system, suchas Gamma function, Bata function, Mittag-Leffer function, etc. Then we introduced thethree basic definitions of fractional calculus,which are the Riemann-Liouville definition,Grunwald was the-Letnikov definition and the Caputo definition,and describe therelationship of the three definitions. We give The basic nature of the fractional calculus.Secondly, when we solving linear fractional-order with multi-state time delaysystem,we consider the given space. We obtained the solution of the system by iterativemethod, and prove the existence and uniqueness of the solution for linearfractional-order with multi-state time delay system, which is homogeneous orinhomogeneous.Thirdly, we used Gamma function, Bata function, Mittag-Leffler function andGeneralized Gronwall’s Inequality to study the finite-time stability analysis offractional-order with multi-state time delay. And then, we obtained necessary andsufficient condition of homogeneous and inhomogeneous fractional order linear systemswith multiple delays. When Multiple delays degradation for a single time delay, we getsolution of some special problem.