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(739)
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where si are complex poles in C \
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and where is a distribution
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Moreover, in the case of a density, the analytical form of is given by: ( ) = 1
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For the proof, the idea is to apply the remainder theorem to function H(s) which is meromorphic in the cut plane C \ The diffusive term then follows naturally from the discontinuity of H on the cut on ; precisely, it is shown that: ( ) = lim+
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In other words, the impulse response h of a fractional differential system breaks up r into a localized part hn of integer order n = i=1 i and a part h of purely diffusive nature We can nd in [DAUP 00, HEL 00] a great number of examples illustrating this decomposition result on some non-standard oscillators 743 Connection with the concept of long memory Finally, let us recall that such systems are said to have long memory in so far as the decrease of the impulse response (in the stable case) is not of exponential type This is determined by a generalized expansion in = 0 of distribution , which is followed by the application of the following lemma LEMMA 71 (Watson) For 1 < 1 < m < m+1 , we have:
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(741) m 1 t1+ m + O(t 1 M )
h (t) =
Thus, by juxtaposing the decomposition result (739), the expression of (740) and the asymptotic analysis (740), the following characterization of stability is obtained THEOREM 72 System (734) is stable in BIBO if and only if the two following conditions are veri ed: in (739), we have e(si ) < 0, for all i; the rst exponent 1 in (740) is strictly positive It should be noted that a priori, si , although a nite number (see [BON 00]), is not known in a simple way in the general case The situation is quite different when the system is more structured, as it emphasized below 744 Particular case of fractional differential systems of commensurate orders The general result given before can then be expressed differently, by using a strong algebraic structure induced by the commensurate character of the derivation orders When saying = s , R is de ned such that H(s) = R( ) It is then enough to decompose the rational fraction R into simple elements ( ) m , to de ne
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by inverse Laplace transform the corresponding basic elements E m ( , t) and to characterize their stability by using an asymptotic analysis similar to the preceding one The function E m ( , t) is the fundamental solution of the operator (D )m ; it belongs to the family of the Mittag-Lef er functions, which is a subset of hypergeometric special functions Using these functions, we obtain the following structure result PROPOSITION 718 We have:
N mn
h(t) =
n=1 m=1
rnm E m ( n , t)
with R( ) =
N n=1
mn m=1 rnm
( n ) m
A re ned asymptotic analysis of the functions E m ( n , t) makes it possible to deduce the following fundamental result for BIBO stability when R = Q/P , with P, Q two coprime polynomials and 0 < < 1 THEOREM 73 We have: BIBO stability | arg | > , 2 C, P ( ) = 0 (742)
In this latter case, the impulse response has the asymptotic: h(t) Kt 1 when t + (743)
NOTE 78 In this case, the poles of the system appearing in decomposition (739) are known analytically; they are exactly sn = n 1/ , but only for those of the preceding n , which verify | arg n | < NOTE 79 In the whole case = 1, we nd with (742) the traditional stability result: absence of poles in the closed right-half plane 75 Example of a fractional partial differential equation An example of propagation phenomenon with long memory, very similar to that with which we will now deal is mentioned in [DAUT 84b] This refers to the original Russian articles [LOK 78a, LOK 78b], but the fundamental difference which exists between the case presented and ours is that the space is unbounded; hence there are no discrete spectra or resonance modes of the physical system; moreover, no relationship with the eigenfunctions of fractional derivation appears