Lyapunov functions for fractional order systems

•New lemma for Caputo fractional derivatives.•Possibility to use quadratics Lyapunov functions in the fractional-order extension of Lyapunov direct method.•Stability analysis of many fractional order nonlinear and/or time-varying systems, scalars and vectorials.

A new lemma for the Caputo fractional derivatives, when 0<α<10<α<1, is proposed in this paper. This result has proved to be useful in order to apply the fractional-order extension of Lyapunov direct method, to demonstrate the stability of many fractional order systems, which can be nonlinear and time varying.