Converse theorems in Lyapunov’s second method and applications for fractional order systems

Abstract

We establish a characterization of the Lyapunov and Mittag-Leffler stability through (fractional) Lyapunovfunctions, by proving converse theorems for Caputo fractional order systems. A hierarchy for the Mittag-Leffler orderconvergence is also proved which shows, in particular, that fractional differential equation with derivation order lesserthan one cannot be exponentially stable. The converse results are then applied to show that if an integer order systemis (exponentially) stable, then its corresponding fractional system, obtained from changing its differentiation order, is(Mittag-Leffler) stable. Hence, available integer order control techniques can be disposed to control nonlinear fractionalsystems. Finally, we provide examples showing how our results improve recent advances published in the specializedliterature.