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corresponding, by Laplace transform, to the symbol: H(s) =
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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
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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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which is mathematically meaningful within a suitable functional framework (see eg [STA 94, MON 98, MAT 08] for technical details; the latter reference making the link with the class of well-posed linear systems) A simple calculation thus shows that the impulse response of the input u-output y system is:
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Its transfer function or its symbol is then, for e(s) > 0:
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EXAMPLE 78 A simple case of a diffusive pseudo-differential operator is that of the fractional integrator I , whose diffusive representation is ( ) = sin for 0 < < 1
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2 Indeed, it is, on the one hand, by completion of this family within a suitable topological framework that we can obtain the space of diffusive pseudo-differential operators and, on the other hand and eventually, these simple systems which are programmed numerically by procedures of standard numerical approximation; see eg [H L 06b]
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We see that one of the advantages of diffusive representations is to transform non-local problems of hereditary nature, in time, into local problems, which speci cally enables a standard and effective numerical approximation (see eg [HEL 00]) On the other hand, when the diffusive representation is positive, the realization suggested has the important property of dissipativity of the pseudo-differential operator (a natural energy functional is then given by E (t) = + ( ) | (t, )|2 d ), which is in this case of the positive type, which has 0 important consequences, particularly for the study of stability coupled systems (see [MON 97] and also [MON 00] for non-linear systems, time-varying, with hysteresis, etc) Now, as far as stability is concerned, it is important to notice that some technicalities must be taken care of in an in nite-dimensional setting (namely, LaSalle s invariance principle does not apply when the pre-compactness of trajectories in the energy space has not been proved a priori: this is the reason why we have to analyze the spectrum of the in nitesimal generator of the semigroup of the augmented system and resort to Arendt-Batty stability theorem, as has been done recently in [MAT 05]) 742 General decomposition result
1 By re-using the notation Y (t) = ( ) t 1 and by strictly limiting ourselves to the + case of strictly proper systems ( L < K ), the following signi cant result is obtained (see [MAT 98a, AUD 00])
THEOREM 71 (D ECOMPOSITION R ESULT) The impulse response h of system (734) of symbol H has the structure: