Analysis, Modeling and Stability of Fractional Order Differential Systems 2

10An Introduction to the Lyapunov Stability of Nonlinear Fractional Order Systems

10.1. Introduction

In automatic control, the main interest of the Lyapunov approach concerns the stability analysis of nonlinear systems. Many works are devoted to this topic, particularly those reported in three famous monographs [SLO 91, KHA 96, SAS 10], which can be considered as references. As it was highlighted previously (see Chapter 8), the Lyapunov stability of nonlinear fractional systems has motivated many conference and journal papers, long before the linear case receives a satisfying solution [MOM 04, FAH 12, WAN 09, CHE 14, LI 14, HUL 15]. A famous paper [LI 09, LI 10], which has been cited in many other papers [SAD 10, FAH 11], is a good example of these publications. It deals with the stability of nonlinear fractional systems, based on the concept of Mittag-Leffler stability. In fact, it is the adaptation to the fractional case of a work by Khalil [KHA 96]. However, as the question of the system initial conditions, obviously fundamental either in the integer order case or the fractional case, relies on Caputo’s definition, this approach is questionable with respect to its generalization.

Therefore, according to our approach in this monograph, we intend to propose an introduction to the Lyapunov stability of nonlinear fractional systems, certainly modest, but motivated by a concern for rigor, based on the main results of the linear case.

As the nonlinear domain is a large topic, ...