P.D.E.s with Discontinuous Hysteresis
The hysteresis relation defined by a delayed relay can be approximated by a sequence of differential inclusions, containing a nonmonotone function and a time relaxation term. The asymptotic behaviour of systems obtained by coupling such a relaxation law w
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Outline The hysteresis relation defined by a delayed relay can be approximated by a sequence of differential inclusions, containing a nonmonotone function and a time relaxation term. The asymptotic behaviour of systems obtained by coupling such a relaxation law with either an D.D.E. or a P.D.E. is considered. The equations
a
-(u
at
+ w) -
Llu
=J
au -L1u+w=J
-
at
in Qx]O,T[, in Qx]O,T[,
(1)
(2)
where Q is an open subset of RN (N ~ I) and f is a given function, are studied, with u and w related by a completed delayed relay operator. Several results of Chaps. IX and X are here extended. Multiplicity of solutions for the problem associated to equation (2) is discussed. A quasilinear first order hyperbolic equation containing a delayed relay operator is then considered. Existence and uniqueness of a solution fulfilling a generalized entropy condition is proved. It is shown that the Preisach model can be obtained by homogenization of a periodic space dependent distributions of delayed relays. A related P.D.E. problem is also considered. A P.D.E. model with discontinuous hysteresis issued from biology is outlined. The main results about P.D.E.s with either continuous or discontinuous memory operators are then summarized. Prerequisites. Some notions of functional analysis are used, in particular knowledge of basic function spaces is required. Some definitions are recalled in Sects. XII. 1 and XIl2. Acquaintance with the methods of analysis of linear and nonlinear partial differential equations in Sobolev spaces is needed in Sects. Xl3 and XI.4. Definitions ofVisintin, fundamental function are recalled in Sects. Xll} and Xn.2. A. Differential Modelsspaces of Hysteresis © Springer-Verlag Berlin Heidelberg 1994
326
XI. P.D.E.s with Discontinuous Hysteresis
The definition and the variational formulations of completed delayed relay operators, given in Sects. Vl.l and V1.2, are used.
XI.1 Genesis of O.D.E.s with Hysteresis Discontinuous hysteresis relations can be approximated by introducing a time relaxation term into an equation containing a nonmonotone function. In this section we couple such a law with an O.D.E., and study the asymptotic behaviour as the relaxation constant vanishes.
Approximation by Time Relaxation. Let us assume that P := (PI, P2) E R2 (PI < U E Co([o, Tn, wO E {-I, I}, and set
P2),
A (v):= P2 - PI V p
=
=
2
_
PI + P2
2
Vv E R.
=
Thus Ap(-l) -P2, Ap(l) -PI (note that if P (-1, 1), then Ap coincides with the identity). We want to study the asymptotic behaviour of the following problem, as the positive parameter e vanishes, {
eWe + 8I[-I,ll(we) :3 we(O)
U
=woo
+ Ap(We)
a.e. in ]0, T[ ,
(1.1)
The inclusion (1.1)1 is equivalent to the following variational inequality -1 ~ We ~ 1 in ]0, T[, { (1.2) [eWe-Ap(We)-u](we-v):::;O VvE[-I,I], a.e. in ]O,T[. This dynamics is outlined in Fig. 1. As we shall see, it approximates the delayed relay operator defined in Sect. IV.I, see (IV.1.26), (IV.1.27).
h;
w
~
....
:"
....
(I! ,
j j
~
~
j
..... .'.. , .'.' p~
,
u
'
j
j
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