Stability of fractional-order nonlinear systems by Lyapunov direct method

In this study, by using a characterisation of functions having a fractional derivative, the authors propose a rigorous fractional Lyapunov function candidate method to analyse the stability of fractional-order nonlinear systems. First, they prove an inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption of the existence of the derivative of pseudo-states. Second, they establish fractional Lyapunov functions to fractional-order systems without the assumption of the global existence of solutions. Their theorems fill the gaps and strengthen results in some existing papers.