Stability properties for generalized fractional differential systems

École Nationale Supérieure des Télécommunications, Département Traitement du Signal et des Images & CNRS, URA 820, 46, rue Barrault, 75 634 Paris Cedex 13, France

Abstract

In the last decades, fractional differential equations have become popular among scientists in order to model various stable physical phenomena with anomalous decay, say that are not of exponential type. Moreover in discrete-time series analysis, so-called fractional ARMA models have been proposed in the literature in order to model stochastic processes, the autocorrelation of which also exhibits an anomalous decay. Both types of models stem from a common property of complex variable functions: namely, multivalued functions and their behaviour in the neighborhood of the branching point, and asymptotic expansions performed along the cut between branching points. This more abstract point of view proves very much useful in order to extend these models by changing the location of the classical branching points (the origin of the complex plane, for continuous-time systems). Hence, stability properties of and modelling issues by generalized fractional differential systems will be adressed in the present paper: systems will be considered both in the time-domain and in the frequency-domain; when necessary a distinction will be made between fractional differential systems of commensurate and incommensurate