We establish a characterization of the Lyapunov and Mittag-Leffler stability through (fractional) Lyapunov functions, by proving converse theorems for Caputo fractional order systems. A hierarchy for the Mittag-Leffler order convergence is also proved which shows, in particular, that fractional differential equation with derivation order lesser than one cannot be exponentially stable. The converse results are then applied to show that if an integer order system is (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 fractional systems. Finally, we provide examples showing how our results improve recent advances published in the specialized literature.