Fattorini H.O., Kerber A. The Cauchy Problem
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8.7. Miscellaneous Comments
507
theorem and the Lax equivalence theorem) have been extended to the
general equation (8.4.1) (see the author [1982: 2]).
Hereditary Differential Equations.
The systems described shortly
after Theorem 8.3.1 do not in general satisfy Huygens' principle; the state of
the system at a single time s < t does not suffice to determine the state at
time t (compare with Section 2.5(b)). That this type of situation is common
in physical phenomena was well known more than a century ago, although
its mathematical modeling had to wait for some time. Hereditary equations
were introduced (and named) by Picard [1907: 1]. Their importance in
physical, and especially biological phenomena was recognized by Volterra
[1909: 11, [1928: 1], [1931: 11, who undertook a systematic study of what
would be called now ordinary integro-differential equations. We quote from
[1931: 1, p. 142]: "L'etat d'un systeme biologique a un moment donne
semble donc bien devoir dependre des rencontres ayant eu lieu pendant une
periode plus on moins longue precedant ce moment; et dans les chapitres
qu'on vient de developper, on a, somme toute, neglige la duree de cette
periode. Il convient de tenir compte maintenant de l'influence du passe."
"On rencontre, en physique,
dans
l'etude
de
1'elasticite,
du
magnetisme, de l'electricite, bien des phenomenes analogues de retard,
trainage ou histeresis. On peut dire que dans le monde inorganique it existe
aussi une memoire du passe, comme la memoire du fil de torsion dont la
deformation actuelle depend des etats anterieurs."
A few lines below, Volterra remarks: "Mais lorsqu'en physique
1'heredite entre en jeu, les equations differentielles et aux derivees partielles
ne peuvent pas suffire; sinon les donnees initiales determineraient 1'avenir.
Pour faire jouer un role a la suite continue des etats anterieurs (infinite de
parametres ayant la puissance du continu) it a fallu recourir a des equations
integrales et integro-differentielles oiz figurent des integrales sous lesquelles
entrent les parametres caracteristiques du systeme fonctions du temps
pendant une periode anterieure a l'instant considers; on a meme introduit
des types plus generaux d'equations aux derivees fonctionnelles."
After these investigations, interest in hereditary differential equations
faded for almost two decades, except for work such as that of Minorski
[ 1942: 11, where delay differential equations x'(t) = f (t, x (t), x (t - h)) are
examined. Beginning in the late forties, an intense revival occurred, stimu-
lated in part by control theory; the objects of study were functional
differential equations, in general of the form
x'(t) = f(t,x1),
(8.7.4)
where for each t, f is a vector-valued functional defined in a suitable
function space of vector-valued "histories" xt = (x(s), - oc < s s t). As-
suming the histories are continuous functions, the functional is continuous,
linear and time independent, and the system has finite memory, one obtains
508
The Cauchy Problem in the Sense of Vector-Valued Distributions
the linear autonomous equation
x'(t)= ° s(ds)x(t+s), (8.7.5)
f' r
where s is a matrix-valued measure of appropriate dimension. For an
excellent treatment of functional differential equations, we refer the reader
to the recent monograph of Hale [1977: 1], where extensive references and
many examples can be found. We point out in passing that there exists an
interesting and important relation between the equation (8.7.5) and the
theory of the abstract Cauchy problem in Chapter 2, which can be roughly
described as follows. Let be, say, a continuously differentiable solution
of (8.7.5) with
x(t)=xo(t) (-r<t<O). (8.7.6)
Consider the function
t- u(t)=(x(t+s); -r<s<0)
defined in t >, 0 with values in the "space of histories" JCr of all (vector-
valued) continuous functions defined in - r < t < 0. Then satisfies the
abstract differential equation
u'(t) = Au(t) (t > 0)
(8.7.7)
in the space E = Jl.r, where A is the differentiation operator
Ax(s) = x'(s) (8.7.8)
with domain consisting of all continuously differentiable functions x(.)
satisfying the "nonlocal boundary condition"
x'(0)- f°µ(ds)x(s)ds=0.
(8.7.9)
r
Conversely, if
is a ACS valued function satisfying (8.7.7), we must have
u(t)(s) = x(t + s) (- r < s < 0) because of the definition of A; thus condi-
tion (8.7.9) guarantees the functional differential equation (8.7.5). This
equivalence was used by Krasovskii, Hale and Shimanov (see Hale [1977: 1,
Ch. 7]) to derive many deep results for the linear autonomous equation, as
well as for their nonlinear counterparts.
The study of functional differential equations in infinite-dimensional
spaces is of very recent data. Some (but not all) of the literature below
makes use of variants of the "reduction to differential equations" outlined
above for the equation (8.7.5). See Artola [1967: 1], [1969: 1], [1969: 2],
[ 1970: 11, Datko [ 1976: 11, [1977: 11, Dickerson-Gibson [ 1976: 11, Hlavacek
[1971: 1], Korenevskii-Fes6enko [1975: 11, R. K. Miller [1975: 1], Monari
[1971: 11, Pachpatte [1976: 11, Slemrod [1976: 1], Travis-Webb [1974: 1],
Webb [ 1974: 1], [1976: 11, Zamanov [ 1967: 11, [1968: 11, [1968: 3], [1969: 11,
and Zaplitnaja [1973: 1].
8.7. Miscellaneous Comments
509
The theory in this chapter formally contains the theory of functional
differential equations of the form (8.7.5), but only in reference to the
Cauchy problem in the sense of distributions. A theory of the Cauchy
problem for (8.4.1) in the strong sense (as that in Chapter 2 for differential
equations) seems to have been developed only in particular instances (some
of which one references quoted above). In most of them P = 8'(& I - 8®A -
P where P, is in some sense subordinate to A. Also, nonlinear equations
are studied in many cases by application of the theory of nonlinear
semigroups. Spaces of histories have been studied, among others, by
Coleman-Mizel [1966: 1].
References
This section lists papers and monographs quoted in the text or closely
related to the contents of the book. Additional references can be
found in Amerio-Prouse [1971: 1], Butzer-Berens [1967: 1], Carroll
[ 1969: 2] and [ 1979: 2], Carroll-Showalter [ 1976: 11, Chernoff [ 1974: 1 ],
Daleckii-Krein [ 1970: 1 ], I ille-Phillips [ 1957: 1 ], Krein [ 1967: 1 ],
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