Attractiveness and stability for Riemann–Liouville fractional systems

We propose a novel approach to study the asymptotic behavior of solutions to Riemann–Liouville (RL) fractional equations. It is shown that the standard Lyapunov approach is not suited and an extension employing two (pseudo) state spaces is needed. Theorems of Lyapunov and LaSalle type for general multi-order (commensurate or non-commensurate) nonlinear RL systems are stated. It is shown that stability and passivity concepts are thus well defined and can be employed in L²-control. Main applications provide convergence conditions for linear time-varying and nonlinear RL systems having the latter a linear part plus a Lipschitz term. Finally, computational realizations of RL systems, as well as relationships with Caputo fractional systems, are proposed.