Stability Properties for Generalized Fractional Differential Systems

In the last decades fractional di erential equations have become popular among scientists in order to model various stable physical phenom ena 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 auto correlation of which also exhibits an anomalous decay Both types of models stem from a common property of complex variable functions namely mul tivalued 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 sta bility properties of and modelling issues by generalized fractional di erential 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 distinc tion will be made between fractional di erential systems of commensurate and incommensurate orders R esum e Ces derni eres ann ees les equations di erentielles fractionnaires ont et e de plus en plus utilis ees par les scienti ques d esireux de mod eliser divers ph enom enes physiques stables mais pr esentant une d ecroissance lente c est a dire qui ne soit pas de type exponentiel D autre part dans le domaine de l analyse des s eries temporelles des mod eles ARMA fractionnaires ont et e pro pos es de fa con a mod eliser des processus stochastiques dont l autocorr elation est aussi a d ecroissance lente Ces deux types de mod eles proviennent d une propri et e commune des fonctions de la variable complexe a savoir les fonc tions multivalu ees et leur comportementau voisinage du point de branchement ainsi que des d eveloppements asymptotiques e ectu es le long de la coupure qui relie les points de branchement Ce point de vue plus abstrait r ev ele toute son utilit e lorsqu on veut etendre ces mod eles en changeant la position des points de branchement classiques l origine du plan complexe pour les syst emes en temps continu Ainsi nous etudierons les propri et es de stabilit e des syst emes di erentiels fractionnaires g en eralis es et les cons equences sur la mod elisation nous consid ererons les syst emes tant dans le domaine temporel que dans le domaine fr equentiel et si n ecessaire nous ferons la distinction entre syst emes di erentiels fractionnaires d ordres commensurables ou incommensurables