Generalized fractional di erential and di erence equations : stability properties and modelling issuesDenis

In the last decades, fractional diierential equations have become more and more popular among scientists and daring engineers 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 (s = 0 for continuous-time systems , or z = 1 for discrete-time systems). Hence, stability properties of and modelling issues by generalized fractional diierential and diierence systems will be considered in the present paper. In the elds of continuous-time modelling, fractional derivatives have proved useful in linear viscoelasticity, acoustics, rheology, polymeric chemistry... For a treatment of so-called fractional diierential equations (FDEs), we refer to 19, chap. 8, sec. 42], 15, chap. 5 & 6] and 8]. There has been some recent advances in control theory of such systems (see e.g. 9] for stability questions, 12] for controllability and observability considerations and 13] The author would like to thank Eric Moulines from enst for fruitful discussions on the subject and for the opportunity to present related works on this topic at a meeting of smai; he would also like to thank Mrs. C ecile Amblard and Mrs. Fatiha Djehaf for their cooperation as postgraduate students. for observer-based controller design), together with interesting applications. Turning to the innnite dimension (i.e. dealing with FPDEs) has been motivated by the example of a wave equation in viscothermal medium (see 11], 7] and 10]). Moreover, an interesting idea of generalized fractional diierential systems appeared in 21] in a stochastic framework; in this approach however, new branching points are deenitely poles. In the elds of discrete-time modelling, the famous paper by Granger and Joyeux 4] has provided a wide variety of so-called long-memory models for describing the autocorrelation of discrete-time stochastic processes in-nance and econometrics, mostly. An interesting idea of generalized fractional diierence systems appeared in 22]; but once again, it must be noted that here new branching …