CONTROLLABILITY AND STABILITY OF FRACTIONAL ORDER DYNAMICAL SYSTEMS

Abstract

This thesis is concerned with the study of qualitative properties of dynamical systems such as controllability, stability, stabilization and synchronization/anti-synchronization with or without delay. There are nine chapters in the thesis. Chapter 1 contains introductory matter and literature survey related to stability of rst order systems, controllability, stability and chaotic synchronization of fractional-order systems. Preliminaries and some basic de nitions are given in Chapter 2, which are required in subsequent chapters. Chapter 3 concerns the development of asymptotic stability and stabilizability of a class of nonlinear dynamical systems with xed delay in state variable. New su cient conditions are established in terms of the system parameters such as the eigenvalues of the linear operator, delay parameter, and bounds on the nonlinear parts. Finally, examples are given to testify the e ectiveness of the proposed theory. In Chapter 4 the stability analysis of a class of fractional order bimodal piecewise nonlinear system is presented. The existence and uniqueness of solution of the system is established by assuming continuity condition involving the state variable and Lipschitz continuity of the nonlinear function with respect to the state variable. Then suitable su - cient conditions for the asymptotic stability of the system has been proposed. Finally, two examples with numerical simulations are given to empirically testify the proposed stability conditions. In Chapter 5, we consider a class of nonlinear fractional-order control system with delay in state variable. Existence and uniqueness of solution is shown by using method of steps. Then the sensitivity of the state is shown with respect to the initial state and perturbed nonlinear function of the system. Finally, numerical examples are given to validate the i analytical results. Chapter 6, deals with the development of synchronization and anti-synchronization of a fractional-order delay nancial system with market con dence by using an active control approach. Firstly, a Gauss-Seidel like predictor-corrector scheme is proposed to solve fractional-order delay systems. Then numerical comparisons of this scheme with the existing two schemes are shown via an example. Furthermore, numerical simulations are given to show that the nancial system has chaotic behaviours for di erent values of time-delay and fractional-order. Then a suitable active control for synchronization/anti-synchronization of the system has been proposed. Finally, the e ectiveness and validity of the proposed control are shown with the help of two numerical simulations for di erent fractional orders and time-delays. In Chapter 7, a class of fractional-order 2 (1; 2] semilinear control systems with delay in Banach space is considered. Su cient condition for exact controllability has been established by using Sadovskii's xed point theorem and the theory of strongly continuous -order cosine family. An example is given to illustrate the result. Chapter 8, is concerned with trajectory controllability of a class of fractional-order 2 (1; 2] semilinear control systems with delay in state variable. The nonlinearity is considered with respect to both state and control variables. Firstly, the existence and uniqueness of the system is proved under suitable conditions on the nonlinear term involving state variable. Then the trajectory controllability of this class of systems is studied using Mittag-Le er functions and Gronwall-Bellman inequality. Finally, examples are given to illustrate the proposed theory. The conclusion of the thesis and possible directions of future work is given