Asymptotic analysis of Riemann-Liouville fractional systems. (English) Zbl 1413.34022
Summary: 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 \(\mathcal{L}^2\)-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.
MSC:
34A08 | Fractional ordinary differential equations |
34D05 | Asymptotic properties of solutions to ordinary differential equations |
34D20 | Stability of solutions to ordinary differential equations |