Fractional calculus of the origins can be traced back to the end of theseventeenth century， the time when Newton and Leibniz developed thefoundations of differential and integral calculus. Nowadays， fractionaldifferential equations have been proved to be an excellent tool in themodelling of many phenomena in various fields of engineering， physics，and economics. Many practical systems can be represented more accuratelythrough fractional derivative formulation. However， time delay is acommon phenomenon in the objective world and engineering fields. Inorder to describe the system more accurately in many practical systems， weneed to take the influence of fractional-order derivative and delay intoconsideration altogether. Therefore， it has a practical significance to studythe solution and its characteristics of fractional delayed differentialsystems.
　　In this book， existence and expression of solution， stability and controlproblem of fractional-order delayed differential systems are discussed. Thisbook， composed of six chapters， mainly studies existence conditions ofsolution of fractional order functional differential equations， and generalsolution of fractional order linear differential difference equations， andstability and control problems of fractional-order delayed differentialsystems.
　　Firstly， the background and significance of the problems are given. We review the development history of fractional calculus. The main work done is also introduced in this book.
　　Secondly， we recall the existence and uniqueness of solutions to the Cauchy type problems for ordinary differential equations of fractional order on a finite interval of the real axis in spaces of continuous functions.Nonlinear and linear fractional differential equations in one-dimensional and vectorial cases are considered. The corresponding results of the Cauchy problem for ordinary differential equations are presented.
　　Thirdly， the existence results of solution of fractional order functional differential equation are derived based on Banach fixed point theorem，Schauder fixed point theorem and successive approximations technique，respectively. These results extend the corresponding ones of ordinary
　　differential equations and functional differential equations ofinteger order.
　　Next， we investigate the general solution of fractional-order linear fractional neutral type differential-difference equations. The exponential
　　estimates of the solutions and the expressions of general solution for linear fractional neutral differential difference equations are derived by using the
　　Gronwallintegral inequality and the Laplace transform method， respectively.At the same time， we also establish the variation of constant formulae for linear time varying Caputo fractional delay differential system.
　　Further， we apply the algebraic approach to discuss the asymptotic stability of fractional-order linear singular delay differential systems. The sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria， one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and efficient.
　　Finally， we discuss the controllability oflinear fractional differential systems with delay in state and impulses. The expression of state response for such systems is derived， and the sufficient and necessary conditions of controllability criteria are established. Both the proposed criteria and illustrative examples show that the controllability property of the linear systems is independent on the order of fractional derivative， or on delay or impulses.
　　The purpose of this book focuses on the existence and expression of solution， stability and control problems of fractional-order delayed differential systems.

　　Fractional calculus of the origins can be traced back to the end of theseventeenth century， the time when Newton and Leibniz developed thefoundations of differential and integral calculus. Nowadays， fractionaldifferential equations have been proved to be an excellent tool in themodelling of many phenomena in various fields of engineering， physics，and economics. Many practical systems can be represented more accuratelythrough fractional derivative formulation. However， time delay is acommon phenomenon in the objective world and engineering fields. Inorder to describe the system more accurately in many practical systems， weneed to take the influence of fractional-order derivative and delay intoconsideration altogether. Therefore， it has a practical significance to studythe solution and its characteristics of fractional delayed differentialsystems.
　　In this book， existence and expression of solution， stability and controlproblem of fractional-order delayed differential systems are discussed. Thisbook， composed of six chapters， mainly studies existence conditions ofsolution of fractional order functional differential equations， and generalsolution of fractional order linear differential difference equations， andstability and control problems of fractional-order delayed differentialsystems.
　　Firstly， the background and significance of the problems are given. We review the development history of fractional calculus. The main work done is also introduced in this book.
　　Secondly， we recall the existence and uniqueness of solutions to the Cauchy type problems for ordinary differential equations of fractional order on a finite interval of the real axis in spaces of continuous functions.Nonlinear and linear fractional differential equations in one-dimensional and vectorial cases are considered. The corresponding results of the Cauchy problem for ordinary differential equations are presented.
　　Thirdly， the existence results of solution of fractional order functional differential equation are derived based on Banach fixed point theorem，Schauder fixed point theorem and successive approximations technique，respectively. These results extend the corresponding ones of ordinary
　　differential equations and functional differential equations ofinteger order.
　　Next， we investigate the general solution of fractional-order linear fractional neutral type differential-difference equations. The exponential
　　estimates of the solutions and the expressions of general solution for linear fractional neutral differential difference equations are derived by using the
　　Gronwallintegral inequality and the Laplace transform method， respectively.At the same time， we also establish the variation of constant formulae for linear time varying Caputo fractional delay differential system.
　　Further， we apply the algebraic approach to discuss the asymptotic stability of fractional-order linear singular delay differential systems. The sufficient conditions are presented to ensure the asymptotic stability for any delay parameter. By applying the stability criteria， one can avoid solving the roots of transcendental equations. The results obtained are computationally flexible and efficient.
　　Finally， we discuss the controllability oflinear fractional differential systems with delay in state and impulses. The expression of state response for such systems is derived， and the sufficient and necessary conditions of controllability criteria are established. Both the proposed criteria and illustrative examples show that the controllability property of the linear systems is independent on the order of fractional derivative， or on delay or impulses.
　　The purpose of this book focuses on the existence and expression of solution， stability and control problems of fractional-order delayed differential systems.

Chapter 1 Introduction
1．1 Fractional Calculus
1．2 Outline of This Book

Chapter 2 Overview of Solutions to Fractional ODEs
2．1 Existence Results of Fractional ODEs on a Finite Interval of the Real Axis
2．2 Global Solutions to ODEs with the Riemann-Liouville Derivative
2．3 Local Solutions to ODEs with the Riemann-Liouville Derivative
2．4 The Cauchy Problems of ODEs with the Caputo Derivative

Chapter 3 Existence of Nonlinear FFDEs
3．1 Problem Formulation
3．2 Definitions and Lemmas
3．3 Existence Results of Fractional-Order FDEs
3．4 An Illustrative Example

Chapter 4 Solutions of Linear Fractional Delayed Differential Equations
4．1 On the Solution of Fractional Delayed Differential Systems
4．2 General Solution of Linear Fractional Neutral DDEs
4．3 Variation of Constant Formulae for Fractional DDEs

Chapter 5 Stability of Fractional Delayed Differential Equations
5．1 Stability of Fractional Linear Singular Delayed Differential Systems
5．2 Stability of Fractional Neutral Dynamical Systems with Multiple Delays
5．3 Finite-Time Stability of Fractional Distributed Delayed Neural Networks

Chapter 6 Controllability of Fractional Delayed Differential Systems
6．1 Controllability Criteria for Linear Fractional Differential Systems with State
Delay andlmpulses
6．2 Reachability and Controllability of Fractional Singular Dynamical Systems
with Control Delay

References