Asymptotical stability of Riemann–Liouville fractional nonlinear systems

Stability analysis of fractional nonlinear differential systems with Riemann–Liouville derivative remains an open problem. This paper is concerned with this open problem, and we first present two new inequalities on Riemann–Liouville fractional derivative, which play important roles in the investigation of the stability. Applying Lyapunov direct method, several sufficient conditions on asymptotical stability of fractional nonlinear systems without and with unbounded delays are presented. The advantage of our employed method is that one may directly calculate integer-order derivative of the Lyapunov function. It is convenient to check stability of practical systems by using our proposed method, and two simple examples are given to show the efficiency of our results.