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NOTE 77 Strictly speaking, the term fractional should be reserved for the commensurate systems of orders ( l = l 1 and k = k 1 ), whereas the term non-integer would be, in truth, more suitable; we conform here to the Anglo-Saxon use (fractional calculus) In section 741 we give a general structure result which shows to what extent the fractional differential systems are also diffusive pseudo-differential systems In section 743, a characterization of the concept of long memory is given Finally, in section 744, we recall the particular case of the fractional differential systems of commensurate orders, to which the general structure result naturally applies, but which allows, moreover, an explicit characterization of stability (in the sense of BIBO) However, rst of all, in section 741, we recall some basic ideas on what diffusive representations of pseudo-differential operators are

741 Introduction to diffusive representations of pseudo-differential operators A rst-order system, or autoregressive lter of order 1 (AR-1) in other contexts, is undoubtedly the simplest linear dynamic system imaginable which does not oscillate, but has a behavior of pure relaxation A discrete superposition of such systems, for various time constants k , or in an equivalent way for various relaxation constants 1 k = k and various weights k , gives a simple idea without being simplistic2 of the diffusive pseudo-differential operators required to simulate the fractional differential equations When the superposition is discrete and nite, the resulting system is a system of a integer order with poles (real negative sk = k ) and of zeros; on the other hand, if the superposition is either discrete in nite, or continuous for all the relaxation constants > 0 and with a weight function ( ), we obtain a pseudo-differential system known to be of the diffusive type, the function being called the diffusive representation of the associated pseudo-differential operator In the sense of systems theory, a realization of such a system will be: t (t, ) = (t, ) + u(t)

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(735) (736)

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