BV -EXTENSION OF
RATE INDEPENDENT OPERATORS
VINCENZO RECUPERO
Abstract. Rate independent operators naturally arise in the mathematical analysis of hysteresis. Among rate independent operators, the locally monotone ones
are those better suited for the study of PDE’s with hysteresis. We prove that a
rate independent operator R : Lip(0, T ) −→ BV (0, T ) ∩ C(0, T ), which is locally
monotone and continuous with respect to the strict topology of BV admits a
e : BV (0, T ) −→ BV (0, T ). This result provides a
unique continuous extension R
general extension theorem which applies to several concrete hysteresis operators.
For many of these operators the existence of a continuous extension was previously
known at most to the space BV (0, T ) ∩ C(0, T ).
1. Introduction
Let T > 0 and let F (0, T ) be a set of functions on the time-interval [0, T ]. A
(nonlinear) operator R : F (0, T ) −→ R[0,T ] is called rate independent if for every
increasing continuous surjective reparametrization φ : [0, T ] −→ [0, T ] the relation
R(u ◦ φ) = R(u) ◦ φ
(1.1)
holds for every u ∈ F (0, T ) such that u ◦ φ ∈ F (0, T ). Rate independent operators
naturally arise in the mathematical analysis of hysteresis phenomena: the operator
R is called a hysteresis operator if (1.1) holds together with the so-called Volterra
property
(1.2)
u = v on [0, t]
=⇒
R(u)(t) = R(v)(t) ∀u, v ∈ F (0, T )
for every t ∈ [0, T ]. The fact that properties (1.1)-(1.2) characterize hysteresis is
clear if one thinks of hysteresis diagrams and loops that appear in ferromagnetism,
elastoplaticity, shape memory alloys, and so on: relation (1.1) allows a graphic
representation of w = R(u) as loops in the plane (u, w), without any reference to
time, i.e., without the need of prescribing the velocity of u and specifying the velocity
of w.
2000 Mathematics Subject Classification. Primary: 47H30,47H99,74N30; Secondary: 26A99.
Key words and phrases. Rate independent operators, hysteresis, extending rate independent
operators, functions of bounded variation of one variable.
Research partially supported by University of Padova (Italy) and by the project “Problemi di
frontiera libera nelle scienze applicate”, 2002-2004, of italian MIUR.
2
VINCENZO RECUPERO
In the last decades the analysis of hysteresis models has received an increasing
attention by pure and applied mathematicians, as the recent monographs [6, 12, 13,
17, 20] show. A big amount of work has also been made to study partial differential
equations involving hysteretic nonlinearities (see [13, 20]). In the case when these
nonlinearities occur in the principal part of the differential operator, the analysis
becomes very nontrivial and it turns out that the only class of hysteresis operators which can be profitably studied is the class of the so-called locally monotone
operators, i.e. those operators R such that
u monotone increasing (resp. decreasing) on [a, b]
=⇒
R(u) monotone increasing (resp. decreasing) on [a, b].
This condition is also natural if we recall that usually hysteresis diagrams occurring
in applications consist of increasing branches: this is exactly the geometrical meaning
of local monotonicity.
Usually a hysteresis operator is defined only on a class of regular functions, especially in engineering works. A common procedure consists in defining the operator
R first on the set of continuous functions which are piecewise monotone. Then R
is extended to all piecewise monotone functions in a canonical way, as shown in [6,
§2.2] and [16]. If the original operator is also uniformly continuous with respect to
uniform topology, then by density it can be extended to C([0, T ]). However in many
concrete examples hysteresis operators are naturally defined on Lipschitz or absolutely continuous functions, but it is not clear if they can be extended to functions
more general than piecewise monotone ones. In particular it is important to know if
an extension of R exists to the space BV (0, T ) of functions with bounded variation
in such a way that it preseves reasonable continuity properties. When we speak of
continuity properties we refer to continuity with respect to the strict metric, which
induces a natural topology on BV (see (2.3), Section 2.1, for the definition). In
general this extension problem is not solvable: examples are known of hysteresis
operators which are continuous on W 1,1 (0, T ) with respect to strict metric, but that
cannot be extended to BV (0, T ) in a continuous manner (see, e.g., [20, p. 136]).
The aim of this note is to investigate when such extension is possible. We consider
a general rate independent operator R : Lip(0, T ) −→ BV (0, T ) ∩ C(0, T ), we assume that R is continuous with respect to the strict metric and we prove that if R
is locally monotone, then it can be continuously extended to all of BV (0, T ) in a
unique way (in fact an assumption weaker that local monotonicity is needed). In
this way we provide a general extension theorem which applies to a large variety of
hysteresis operators, whereas in the literature every single operator is treated with
ad hoc arguments and in most cases the extension is proved at most only to the
space BV (0, T ) ∩ C([0, T ]). From our theorem we also deduce an extension theorem
for the unbounded case T = ∞.
RATE INDEPENDENT OPERATORS
3
Here is a brief plan of the paper. In the following section we recall the basic facts
about functions of bounded variation of one variable and about rate independent
and hysteresis operators. In section 3 we state the main results of the paper and
in section 4 we present all the proofs. In section 5 we show how our results apply
to several classes of hysteresis operators. Finally in the appendix we recall some
properties of convergence of measures needed in our proofs.
2. Preliminaries
In this section we recall the basic definitions and properties of strict convergence
and rate independent operators. We fix I := ]a, b[, the open interval with endpoints
a, b ∈ [−∞, ∞], a < b, and we deal exclusively with functions in RI , the set of real
functions defined on I. In the sequel N is the set of strictly positive integers {1, 2, . . .}
and L1 is the Lebesgue measure on R. In general we do not identify two functions
agreeing L1 -a.e.. If f ∈ RI is given, Lip(f ) denotes its Lipschitz constant, whereas
Discont(f ) represents the set of discontinuities of f and Cont(f ) := I rDiscont(f ).
We say that f is increasing if (f (t1 ) − f (t2 ))(t1 − t2 ) ≥ 0 for every t1 , t2 ∈ I.
2.1. Functions of bounded variation. In order to state the main properties of
strict convergence of functions of bounded variation, we need some notions about
convergence of measures. The reader is referred to the Appendix for a brief summary.
We adopt the notation of [2, §3.2] to which we refer for the omitted proofs.
Definition 2.1. Let u ∈ RI . We say that u is of bounded variation on I if u ∈ L1 (I)
and its distributional derivative is a finite signed Borel measure, i.e. there exists a
(unique) measure µ =: Du ∈ Mb (I) such that
Z
Z b
ϕ dDu = −
ϕ0 (t)u(t) dt
∀ϕ ∈ Cc1 (I).
I
a
The vector space of all functions of bounded variation on I is denoted by BV (I).
Let J be an interval contained in I. We recall that a subdivision of J is a finite
family of points (tj )m
j=0 , m ∈ N, with the property that t0 < t1 < · · · < tm and
tj ∈ J for j = 0, 1, . . . , m. The pointwise variation pV(u, J) of u on J is defined as
( m
)
X
sup
|u(tj ) − u(tj−1 )| : m ∈ N, (tj )m
j=0 subdivision of J .
j=1
The essential variation eV(u, J) of u on J is defined by
eV(u, J) = inf pV(w, J) : u = w L1 -a.e. in J .
If v ∈ RI is such that pV(v, I) = eV(u, I) < ∞, then v is called good representative
of the L1 -class of u.
4
VINCENZO RECUPERO
If pV(u, I) < ∞, it is well known that u is bounded, Discont(u) is at most
countable, and u admits left and/or right limits u(t−), u(t+) at every limit point t
of I.
Proposition 2.1. Let u ∈ L1 (I). Then u ∈ BV (I) if and only if eV(u, I) < ∞.
In this case there exists a good representative v ∈ RI of u, therefore for every
 
Du
(A) and in particular
open interval A ⊆ I we have pV(v, A) = eV(u, A) = 
eV(u, I) = k Duk. Moreover there exists a unique constant C ∈ R such that the
functions ur , ul ∈ RI defined by
(2.1)
ul (t) := C + Du(]a, t[),
ur (t) := C + Du(]a, t]),
t∈I
are good representatives; ur is right-continuous, ul is left-continuous. Any other
good representative v of u is characterized by the condition
(2.2)
v(t) ∈ (1 − λ)ul (t) + λur (t) : λ ∈ [0, 1]
∀t ∈ I;


Du
({t}) 6= 0} and the constant C appearing in (2.1)
moreover Discont(v) = {t : 
is equal to v(a+).
For a proof of the previous proposition we refer to [2, Theorems 3.27, 3.28]. We
will use the space W 1,p (I), p ∈ [1, ∞], of functions in LRp (I) having distributional
derivatives in Lp (I): if u ∈ W 1,1 (I) we have eV(u, I) = I |u0 (t)| dt. We also recall
that u ∈ W 1,∞ (I) if and only if u has a representative v ∈ L∞ (I) ∩ Lip(I), where
Lip(I) is the space of functions on I with finite Lipschitz constant (see [1, 8, 21] for
these facts).
Convention 2.1. If u ∈ BV (I) we will adopt the convention u(a) := v(a+) and
u(b) := v(b−), where v is any good representative of u. The standard notation I
denotes the closure of I, so that, e.g., R = R. Moreover by the notation C k ([a, b]),
k = 0, 1, . . . , ∞, we indicate the space of k-times differentiable functions f on I for
which f (h) is bounded and uniformly continuous on I for 0 ≤ h ≤ k. With this
convention we have, e.g., that C 1 ([−∞, ∞]) is the set of functions f : R −→ R such
that f and f 0 are bounded and uniformly continuous on R.
With the convention just adopted, from (2.1) it is immediately inferred that
Du(I) = u(b) − u(a), whenever u ∈ BV (I). Using this fact together with a standard
application of Fubini theorem, one finds the following
R
Proposition 2.2. If u ∈ BV (I) and ϕ ∈ C 1 ([a, b]) then I ϕ dDu = ϕ(b)u(b)
Rb
−ϕ(a)u(a) − a ϕ0 u dL1 .
The strict (semi)metric ds on BV (I) is defined by
(2.3)
ds (u, v) := ku − vkL1 (I) + k Duk − k Dvk ,
u, v ∈ BV (I).
A sequence of functions which converges with respect to the strict metric is called
strictly convergent on I. This metric is not complete. We will use the notation
RATE INDEPENDENT OPERATORS
5
BVloc ([a, b[) to denote the set of functions u ∈ L1loc ([a, b[) such that eV(u, ]a, c[) < ∞
for every c ∈ ]a, b[. In this connection, a sequence (un ) in BVloc ([a, b[) is called locally
strictly convergent to u ∈ BVloc ([a, b[) if un → u in L1loc ([a, b[) and eV(un , ]a, c[) →
eV(u, ]a, c[) for every c ∈ I. The strict metric induces the “natural” topology
on BV (I) by virtue of the following approximation result, which is proved using
regularization by convolution (cf., e.g., [21, Theorem 5.3.3]).
Proposition 2.3. If u ∈ BV (I) then there exists a sequence un ∈ C ∞ (I) ∩ BV (I)
which strictly converges to u.
∗
It is easy to see (cf. [2, Proposition 3.13]) that if un → u strictly then Dun * Du
(see Definition 6.1 in the Appendix). Then using Proposition 6.2 of the Appendix
we obtain
Proposition 2.4. Let u, un ∈ BV (I) for every n ∈ N be such that un → u strictly
 
N
N
Dun
*
on I. Then we have the following narrow convergences: Dun * Du and 



Du
 (see Definition 6.1 in the Appendix).
Lemma 2.1. If u, un ∈ BV (I) such that un → u strictly. Then un (a) → u(a) and
un (b) → u(b).
Proof. It suffices to prove the claim for a. Let ϕ ∈ C 1 ([a, b]) be such that ϕ(a) = 1
R
Rb
and ϕ(b) = 0. Then thanks to Proposition 2.2 we have I ϕ dDun = − a ϕ0 un dL1 −
un (a) for every n ∈ N. Therefore we get the thesis taking the limit as n % ∞ and
using Proposition 2.4.
2.2. Rate independent and hysteresis operators. Now we introduce the notions of rate independent operator and of hysteresis operator. In the last decades,
operators of this kind have been extensively studied in several research articles and
in the monographs [6, 13, 12, 17, 20]. We refer to these books for concrete examples.
However in Section 5 we recall the definitions of several hysteresis operators and we
apply to them our results, obtaining a number of new properties.
Definition 2.2. Let F (I) ⊆ RI , F (I) 6= ∅, and let R : F (I) −→ RI .
a) The operator R is called rate independent if
R(u ◦ φ) = R(u) ◦ φ
for every u ∈ F (I) and for every φ : I −→ I continuous, increasing, surjective
such that u ◦ φ ∈ F (I).
b) We say that R has the Volterra property (or is causal ) if
u = v on ]a, t]
=⇒
R(u)(t) = R(v)(t).
for every u, v ∈ F (I) and for every t ∈ I.
c) If R is rate independent and has the Volterra property, then it is called
hysteresis operator.
6
VINCENZO RECUPERO
In most of the literature on this subject, the interval under considerations is
I = ]0, T [, where T > 0 is a finite time horizon. We gave the previous definitions
in a more general context because the unbounded case might be interesting for the
study of asymptotic properties of systems with hysteresis (this is also done in [16]).
Notice that in defining φ from I into itself, we allow, e.g., time rescalings that are
equal to b ∈ R on an interval ]t0 , b[ for a certain time t0 ∈ ]a, b[.
Definition 2.3. Let F (I) ⊆ RI , F (I) 6= ∅, and let R : F (I) −→ RI .
a) We say that R is locally monotone if for every c, d ∈ I, c < d,
u monotone increasing (resp. decreasing) on [c, d]
=⇒
R(u) monotone increasing (resp. decreasing) on [c, d].
b) We say that R is locally isotone if for every c, d ∈ I, c < d,
u monotone on [c, d]
=⇒
R(u) monotone on [c, d].
The class of locally monotone operator contains the most significant hysteresis
operators: in fact under natural assumptions all the most common hysteresis operators are locally monotone. However we need a weaker assumption, namely the local
isotony, introduced in part b) of Definition 2.3. We warn the reader that we have
not found the notion of local isotony in the literature.
3. Main results
Within this section we state the main results of the paper. We need some preliminary notions. Following [11, §2.5.16], we define a sort of arc lenght of a function
of bounded variation. The situation here is slightly different because we want two
functions which are equal L1 -a.e. to have the same arc lenght; moreover a normalization factor is needed. Recall that I = ]a, b[ and assume now that a, b are finite
and u ∈ BV (I). We define the normalized arc lenght function `u : I −→ I by setting
(



(]a, t[) if k Duk 6= 0
Du
a + kb−a
Duk
,
t ∈ I.
(3.1)
`u (t) :=
a
if k Duk = 0
The function `u is increasing, Discont(`u ) = Discont(ul ), and
[
`u (I) = I r
]`u (t−), `u (t+)] .
t∈Discont(ul )
If t1 < t2 we have |ul (t1 ) − ul (t2 )| ≤ pV(ul , [t1 , t2 ]) = pV(ul , ]a, t2 ]) − pV(ul , ]a, t1 ]) =
eV(u, ]a, t2 [) − eV(u, ]a, t1 [) therefore
k Duk
|`u (t1 ) − `u (t2 )|
∀t1 , t2 ∈ I.
b−a
This inequality yields that ul (`−1
u (σ)) is a singleton for every σ ∈ `u (I), therefore
there is a unique function U : `u (I) −→ R such that U ◦ ` = ul . From (3.2) it also
(3.2)
|ul (t1 ) − ul (t2 )| ≤
RATE INDEPENDENT OPERATORS
7
follows that U is the unique Lipschitz function such that U ◦`u = u L1 -a.e.. Moreover
Lip(U ) ≤ k Duk/(b − a). In order to extend U to all of I we define u
e : I −→ R by
setting
u
e(σ) := (1 − λ)ul (t) + λul (t+)
if σ = (1 − λ)`u (t) + λ`u (t+), t ∈ I, λ ∈ [0, 1].
It is clear that u
e extends U and that Lip(e
u) = Lip(U ). The function u
e may be
regarded as a kind of reparametrization of u by the normalized arc lenght. We will
prove the following result:
Proposition 3.1. Assume a, b are finite and let u ∈ BV (I). Let `u : I −→ I be
its “normalized” arc lenght defined by (3.1). Then there exists a unique function
u
e ∈ Lip(I) such that
u=u
e ◦ `u
(3.3)
(3.4)
L1 -a.e. in I,
u
e is affine on [`u (t−), `u (t+)]
∀t ∈ Discont(`u ).
Moreover pV(e
u, I) = eV(u, I) and Lip(e
u) ≤ k Duk/(b − a). Finally if φ : I −→ I is
continuous, increasing, surjective, and v := u ◦ φ, then `v = `u ◦ φ and ve = u
e, or in
]
other terms u ◦ φ = u
e.
Now we are able to state the main theorem.
Theorem 3.1. Assume a, b are finite and let F (I) ⊆ RI be such that Lip(I) ⊆
F (I) ⊆ BV (I) ∩ C(I). Let R : F (I) −→ BV (I) be a rate independent operator
which is locally isotone, continuous with respect to the strict metric and such that
R(Lip(I)) ⊆ C(I). For every u ∈ BV (I) let `u and u
e be the functions given by
Proposition 3.1. Define
e
(3.5)
R(u)
:= R(e
u) ◦ `u .
e : BV (I) −→ BV (I) which
Formula (3.5) defines a rate independent operator R
extends R and which is continuous with respect to the strict metric. If BV (I) is
e is the only continuous extension
regarded a space of L1 -classes of functions, then R
e
of R with respect to the strict topology. Finally if R is a hysteresis operator, then R
is also a hysteresis operator.
In section 5.3 an example shows that the assumption of local isotony cannot be
dropped. In the unbounded case we have the following result:
Theorem 3.2. Assume a is finite and let F (I) ⊆ RI be such that Lip(I) ⊆ F (I) ⊆
BVloc ([a, b[) ∩ C(I). Let R : F (I) −→ BVloc ([a, b[) be a hysteresis operator which is
locally isotone, continuous with respect to the local strict convergence, i.e.
vn → v locally strictly on ]a, c[ ∀c ∈ I
=⇒
=⇒
R(vn ) → R(v) locally strictly on ]a, c[ ∀c ∈ I,
8
VINCENZO RECUPERO
and such that R(Lip(I)) ⊆ C(I). Then, identifying functions which are equal a.e.,
e : BVloc ([a, b[) −→ BVloc ([a, b[) which extends R and that is
there exists a unique R
e is a hysteresis
continuous with respect to the local strict convergence. Moreover R
operator.
4. Proofs
We need some preparatory lemmas. We recall that I = ]a, b[.
Lemma 4.1. Let vn : I −→ R be a sequence of increasing functions which is pointwise converging to a continuous function v : I −→ R. Assume that the sequences
vn (a+) and vn (b−) have a finite limit. Then vn converges uniformly to v.
Proof. If I is bounded, a proof can be found in [9, Theorem 10, p. 166]. If I is
unbounded, the lemma can be easily inferred by the bounded case, e.g., letting ψ be
a homeomorphism from [a, b] to [0, 1] and defining w := v ◦ ψ −1 , wn (s) := vn ◦ ψ −1 ,
s ∈ [0, 1], for every n ∈ N. Then the assumptions on vn (a+) and vn (b−) allow to
apply [9, Theorem 10, p. 166] to wn , and this yields the result for vn .
Lemma 4.2. Let u ∈ BV (I) and let (un ) be a sequence in BV (I) that strictly
converges to u. Then
 
 
(]a, t[) and Dun (]a, t[) → Du(]a, t[) as n % ∞ for
Du
Dun
(]a, t[) → 
(i) 
l
every t ∈ Cont(u );
(ii) un (t) → u(t) for L1 -a.e. t ∈ I;
(iii) if in addition u and un are continuous for every n ∈ N, then un converges
uniformly on I.
 

N 
N

Dun
*
Du
 and Dun *
Proof. By Proposition 2.4 
Du, therefore Proposition








6.2 of the Appendix yields that Dun(]a, t[) → Du(]a, t[) and Dun (]a, t[) →
 
Du
({t}) = 0, i.e. for every t ∈ Cont(ul ). So
Du(]a, t[) for every t ∈ I such that 
the first part of the Lemma is proved. Concerning statement (ii), let us consider the
T
set E := Cont(ul ) ∩ n Cont(uln ) whose complement is at most countable. Hence
we have
(4.1)
un (t) = un (a) + Dun (]a, t[),
u(t) = u(a) + Du(]a, t[)
∀t ∈ E.
Passing to the limit for n % ∞ and using (i) and Lemma 2.1, we find un (t) → u(t) if
t ∈ E. Let us now demonstrate part (iii). It can be deduced using the decomposition
of un as difference of two increasing functions:
un (t) := [un (t) + pV(un , ]a, t])]/2 + [un (t) − pV(an , ]0, t])]/2.
If E is defined as above, the continuity assumption yields E = I, hence the equalities
 
Dun
(]a, t[)
in (4.1) hold everywhere. For the same reason we have pV(un , ]a, t]) = 
RATE INDEPENDENT OPERATORS
9
 
Du
(]a, t[). Thus by (i) and (ii) we find that for every t ∈ I
and pV(u, ]a, t]) = 
 
 
Du
(]a, t[)]/2 + [u(t) − 
Du
(]a, t[)]/2
lim un (t) = [u(t) + 
n%∞
= [u(t) + pV(u, ]a, t])]/2 + [u(t) − pV(u, ]a, t])]/2 = u(t).
Since for any n ∈ N the functions t 7−→ [un (t) + pV(un , ]a, t])]/2 and t 7−→ [un (t) −
pV(un , ]a, t])]/2 are increasing, by Proposition 4.1 this convergence is uniform. Lemma 4.3. Let v : I −→ R be such that pV(v, I) < ∞ and let α : I −→ I be an
increasing function satisfying α(a) = a, α(b) = b, and Discont(v) ∩ Discont(α) = ∅.
Moreover assume that
(4.2)
v monotone on [α(t−), α(t+)]
∀t ∈ Discont(α).
Then pV(v ◦ α, I) = pV(v, I).
Proof. We prove the lemma when α is left continuous, the other cases being similar
(however we need only this case). Recall also that with our convention α(a) =
α(a+) (for a finite), but the Lemma is true also without this restriction. The
inequality pV(v ◦ α, I) ≤ pV(v, I) is obvious, hence pV(v, I) is an upper bound for
P
{ nj=1 | v(α(tj )) − v(α(tj−1 ))| : n ∈ N, a < t0 ≤ · · · ≤ tn < b}. Let ε > 0 be
arbitrarily fixed. There exists a subdivision (tj )nj=0 of I such that
(4.3)
pV(v, I) <
n
X
|v(tj ) − v(tj−1 )| + ε/2.
j=1
For every σ ∈ Discont(α) there is a possibly empty subset Aσ ⊆ {tj } contained in
[α(σ−), α(σ+)[. Adding the points α(σ−) = α(σ), α(σ+) to Aσ , the sum in (4.3)
can only increase. Moreover, thanks to the assumption (4.2) we can also replace Aσ
by {α(σ), α(σ+)} without affecting such a sum. Therefore we can assume that (4.3)
holds for a subdivision (tj ) such that
{tj }nj=0 = {s10 , . . . , s1k1 −1 } ∪ {α(σ1 ), α(σ1 +)} ∪ {s20 , . . . , s2k2 −1 } ∪ {α(σ2 ), α(σ2 +)}∪
m
m+1
· · · · · · ∪ {sm
, . . . , sm+1
0 , . . . , skm −1 } ∪ {α(σm ), α(σm +)} ∪ {s0
km+1 }
where
sikm := α(σi )
σi ∈ Discont(α),
{si0 , . . . , siki } ⊆ α(I)
∀i = 1, . . . , m;
∀i = 1, . . . , m + 1.
Hence, setting
τhi := α−1 (sih )
i = 1, . . . , m + 1,
j = 0, . . . , km+1 ,
10
VINCENZO RECUPERO
we can write (α is left continuous)
n
X
|v(tj ) − v(tj−1 )|
j=1
=
m
X
ki
X
i=1
h=1
i
))| + |v(α(σi +)) − v(α(σi ))|
|v(α(τhi )) − v(α(τh−1
km+1
!
+ |v(α(τ0i+1 )) − v(α(σi +))|
+
X
m+1
|v(α(τhm+1 )) − v(α(τh−1
))|
h=1
The fact that Discont(v) ∩ Discont(α) = ∅ yields that for every i = 1, . . . , m there
exists σ̃i very near σi , such that σi < σ̃i and |v(α(σi +)) − v(α(σ̃i ))| < ε/(2m), so
that
n
X
|v(tj ) − v(tj−1 )|
j=1
≤
m
X
ki
X
i=1
h=1
i
|v(α(τhi )) − v(α(τh−1
)| + |v(α(σ̃i )) − v(α(σi ))|
km+1
!
+
|v(α(τ0i+1 ))
− v(α(σ̃i ))| + ε/m
+
X
m+1
|v(α(τhm+1 )) − v(α(τh−1
)|.
h=1
That is, we have found a subdivision (θj )rj=0 satisfying
pV(v, I) <
n
X
|v(α(θj )) − v(α(θj−1 ))| + ε,
j=1
and the lemma is proved.
Proof of Proposition 3.1. It is clear that u
e is the unique Lipschitz function satisfying
(3.3)-(3.4). From (3.2) we deduce the estimate for its Lipschitz constant. The
equality pV(e
u, I) = eV(u, I) is a consequence of Lemma 4.3 which yields pV(e
u, I) =
pV(ul , I) = eV(u, I). In order to prove the last statement let us observe that
the assumptions on φ yield eV(v, ]0, t[) = pV(ul ◦ φ, ]0, t[) = pV(ul , ]0, φ(t)[) =
eV(u, ]0, φ(t)[) for every t ∈ I, therefore
`v (t) =
b−a
b−a
eV(v, ]0, t[) =
eV(u, ]0, φ(t)[) = (`u ◦ φ)(t)
k Dvk
k Dvk
∀t ∈ I.
Thus we have ve ◦ `v = v = u ◦ φ = u
e ◦ `u ◦ φ = u
e ◦ `v and the thesis follows from the
uniqueness of ve.
For the sake of clarity we recall in a remark a simple fact we will need in the proof
of the following Proposition 4.1.
RATE INDEPENDENT OPERATORS
11
Remark 4.1. Let c, d, wc , wd ∈ R be such that c < d. The function w : [c, d] −→ R
defined by w(t) := wc + (wd − wc )t/(d − c) is the only solution of the boundary
problem u00 = 0, v(c) = wc , v(d) = wd , therefore it is the unique minimizer in {v ∈
W 1,2 (]c, d[) : v(c) = wc , v(d) = wd } of the energy functional v 7−→ kv 0 k2L2 (]c,d[) /2
(cf., e.g., [8, Ch. 8]).
The following proposition relates reparametrizations u
e with strict convergence.
Proposition 4.1. Assume a, b are finite and let u, un ∈ BV (I) for every n ∈ N be
such that un → u strictly. Let ` and `n be the “normalized” arc lenght functions
of u and un defined as in (3.1), and let u
e and u
en be the unique Lipschitz functions
satisfying (3.3)-(3.4) with u, u
e, `u replaced respectively by u, u
e, ` and un , u
en , `n , as
given by Proposition 3.1. Then
`n (t) → `(t)
(4.4)
u
en → u
e
(4.5)
∀t ∈ Cont(ul ),
strictly on I.
Proof. Convergence (4.4) follows from Lemma 4.2-(i). Since, by Proposition 3.1,
Lip(e
un ) is bounded, we have that, at least for a subsequence not relabeled, u
en
converges uniformly to a Lipschitz function u
b. Observe now that, by the uniform
convergence and by (4.4), un (t) = u
en (`n (t)) → u
b(`(t)) for almost every t ∈ I. On
the other hand by Lemma 4.2-(ii) we know that un converges L1 -a.e. to u, hence
u(t) = u
b(`(t)) for a.e. t ∈ I. This means that u
b=u
e on `(I), by construction of
u
e. Now let us show that u
b is affine on the intervals [`(t−), `(t+)], t ∈ Discont(`).
Indeed, since u
e is affine on such intervals, from Remark 4.1 it follows that u
e minimizes
0
the norm kv kL2 (I) among the set of all functions v ∈ Lip(I) such that u = v ◦ `. On
the other hand, as Lip(e
un ) ≤ k Dun k/(b − a), we have
Z b
0 2
ke
un kL2 (]a,b[) ≤
k Dun k2 /(b − a)2 ds = k Dun k2 /(b − a),
a
u
e0n
2
hence
is bounded in L (I), thanks to the strict convergence of un . Therefore, at
least for a subsequence, which we do not relabel, u
e0n converges weakly star to u
b0 in
L2 (I), and, using also Proposition 3.1 and Hölder inequality,
u0n k2L2 (]a,b[) ≤ lim inf k Dun k2 /(b − a)
kb
u0 k2L2 (]a,b[) ≤ lim inf ke
n→∞
2
n→∞
2
= k Duk /(b − a) = k De
uk /(b − a)
Z
b
2
=
|e
u0 (s)| ds /(b − a) ≤ ke
u0 k2L2 (]a,b[) .
a
Thus u
b has the minimum property stated before and, by (3.4), we infer that u
e=u
b
1
on I, so that u
en → u
e uniformly and hence in L (I). Finally exploiting again
Proposition 3.1 we see that k De
un k = k Dun k → k Duk = k De
uk, thus convergence
(4.5) is completely proved.
12
VINCENZO RECUPERO
e maps BV (I) into itself. If u ∈ F (I) then
Proof of Theorem 3.1. It is obvious that R
e
u and `u are continuous, therefore by rate independence we have R(u)
= R(e
u) ◦
e extends R. Proving that R
e is rate independent is
`u = R(e
u ◦ `u ) = R(u) so that R
simple thanks to the last statement of Proposition 3.1, indeed if u ∈ BV (I) and
φ : I −→ I is continuous, increasing, and surjective, then v := u ◦ φ ∈ BV (I) and
e ◦ φ) = R(u]
e
R(u
◦ φ) ◦ `v = R(e
u) ◦ `u ◦ φ = R(u)
◦ φ. To prove continuity let us take
a sequence (un ) which strictly converges to u and let us denote the normalized arc
lenght function `un simply by `n . We start by showing that
e n ) → R(u)
e
R(u
(4.6)
in L1 (I)
as n → ∞. Since R is continuous, we have that R(e
un ) → R(e
u) strictly in BV (I) and
Lemma 4.2-(iii). yields that R(e
un ) → R(e
u) uniformly on I, because R(e
un ) and R(e
u)
are continuous. Hence we have that for a.e. t ∈ I
e n )(t) = R(e
e
R(u
un )(`n (t)) → R(e
u)(`u (t)) = R(u)(t)
when n goes to infinity. Observe also that by the uniform convergence we get
e n )k∞ = sup kR(e
sup kR(u
un ) ◦ `n k∞ ≤ sup kR(e
un )k∞ < ∞,
n∈N
n∈N
n∈N
thus in order to obtain (4.6) it suffices to apply the dominated convergence theorem.
It remains to prove that
(4.7)
e n ))k → k D(R(u))k.
e
k D(R(u
Since R(e
u) is continuous and `u is increasing, R(e
u) ◦ `u is a good representative (see
(2.2)), therefore
e
k D(R(u))k
= k D(R(e
u ◦ `u ))k = pV(R(e
u) ◦ `u , I).
By construction u
e is affine (hence monotone) on the interval [`u (t−), `u (t+)] for
every t ∈ Discont(`u ), therefore R(e
u) is also monotone on these intervals, because R
is locally isotone. Therefore Lemma 4.3 implies that pV(R(e
u) ◦ `u , I) = pV(R(e
u), I)
and we can deduce the equality
(4.8)
e
k D(R(u))k
= k D(R(e
u))k.
The same argument shows that
(4.9)
e n ))k = k D(R(e
k D(R(u
un ))k
for every n ∈ N. Since Lip(I) ⊆ F (I), we have that u
e, u
en ∈ F (I) for every n ∈ N,
hence by the continuity of R on F (I) and by (4.5) we find that
(4.10)
k D(R(e
un ))k → k D(R(e
u))k.
e is proved. If we consider
Collecting (4.8)–(4.10) we get (4.7) and the continuity of R
1
BV (I) as a space of L -classes of equivalence, then the strict metric induces a
Hausdorff topology, therefore the uniqueness of the extension is a consequence of
RATE INDEPENDENT OPERATORS
13
e has the Volterra
the density of Lip(I) in BV (I). It is straightforward to prove that R
property, provided R is a hysteresis operator. Indeed if u = v on [0, t] for some t ∈ I,
then we have `u |[0,t] = `v |[0,t] and hence u
e = ve on [0, `u (t)] = [0, `v (t)]. Therefore
e
e
R(u)(t)
= R(e
u)(`u (t)) = R(e
v )(`u (t)) = R(e
v )(`v (t)) = R(v)(t).
Proof of Theorem 3.2. Let us introduce for every c ∈ I the set
F (]a, c[) := {v c ∈ R ]a,c[ : ∃v ∈ F (I), v| ]a,c[ = v c }.
It turns out that Lip(]a, c[) ⊆ F (]a, c[) ⊆ BV (]a, c[) ∩ C(]a, c[). Then let Rc :
F (]a, c[) −→ R ]a,c[ be the operator defined by setting for every uc ∈ F (]a, c[)
Rc (uc ) := (R(u))| ]a,c[
where u ∈ F (I) is such that u| ]a,c[ = uc .
This is a good definition because R has the Volterra property. For the same reason
Rc maps F (]a, c[) into itself. Let us show that Rc is rate independent. Take uc ∈
F (]a, c[) and φc : [a, c] −→ [a, c] continuous, increasing, and surjective. Then let
u ∈ F (I) be such that u| ]a,c[ = uc and let φ : I −→ I defined by φ(t) := φc (t) for
t ∈ ]a, c[ and φ(t) := t if t ≥ c. Then uc ◦ φc = (u ◦ φ)| ]a,c[ , hence Rc (uc ◦ φc ) =
R(u◦φ) = R(u)◦φ = Rc (uc )◦φc . Now we check that Rc is continuous with respect to
the strict metric. Indeed if ucn → uc strictly on ]a, c[ then define un ∈ F (I) by setting
un := ucn on ]a, c[ and un (t) := ucn (c−) if t ≥ c. Analogously we define a function
u ∈ F (I) starting from uc . Thus un → u locally strictly on I and, by assumption,
R(un ) → R(u) locally strictly on I, and this implies that Rc (ucn ) → Rc (uc ) strictly.
Now we can apply Theorem 3.1 and deduce that for every c ∈ I there exists a unique
g
c ) : BV (]a, c[) −→ BV (]a, c[), continuous for the strict topology. Let
extension (R
e : BVloc ([a, b[) −→ BVloc ([a, b[) setting for every u ∈ F (I) and for every
us define R
t∈I
g
c )(u)(t),
e
R(u)(t)
:= (R
where c > t.
e is well defined thanks to the Volterra property and it is a hysteresis
The operator R
operator. Moreover it is locally strictly continuous by construction. Let us verify
e be another extension of R
that it is the unique extension having this property. Let Q
and u ∈ BVloc ([a, b[). For every c ∈ I there exists ucn ∈ C ∞ ([a, c]) such that ucn → u
strictly on ]a, c[ (cf. Proposition 2.3). If un is the continuation of ucn equal to ucn (c−)
e n ) → Q(u)
e
e
on [c, b[, then un → u locally strictly on I, so Q(u
and R(un ) → R(u)
in
1
e
e
e
L (]a, c[). But Q(un ) = R(un ) for every n ∈ N therefore Q(u) = R(u) a.e. in ]a, c[
e
e
for every c ∈ I, then Q(u)
= R(u).
Remark 4.2. Theorem 2.1 can be extended to the vectorial case, more precisely
we can consider rate independent operators defined on a set F (I; H) such that
Lip(I; H) ⊆ F (I; H) ⊆ BV (I; H) ∩ C(I; H), where H is a Hilbert space. The
theorem still holds provided the following generalization of locally isotone operator
14
VINCENZO RECUPERO
is given: we say that R : F (I; H) −→ H I is locally isotone if for every c, d ∈ I,
c < d,
pV(u, [c, d]) = ku(d) − u(c)kH =⇒ pV(R(u), [c, d]) = kR(u)(d) − R(u)(c)kH .
An extension result in the framework of vector valued functions is obtained in [15, 5]
for a particular hysteresis operator: the so called vectorial play operator (see section
5.1 for the scalar case).
5. Applications
In this section we show that Theorem 3.1 applies to several hysteresis operators
that occur in applications, thereby providing a number of new extension results. In
all the examples I = ]0, T [, where T ∈ ]0, ∞[ and we identify functions which are
equal a.e. on I.
5.1. The play operator. Let γi : R −→ [−∞, +∞] be continuous and increasing
for i = 1, 2 and such that γ1 ≤ γ2 . Let e0 ∈ [γ1 (u(0)), γ2 (u(0))]. For every u ∈
W 1,p (I), p ∈ [1, ∞], let w ∈ W 1,p (I) be the only solution of the variational inequality



w(t) ∈ [γ1 (u(t)), γ2 (u(t))] for a.e. t ∈ I
w0 (t)(w(t) − v(t)) ≤ 0 ∀v ∈ RI , v(t) ∈ [γ1 (u(t)), γ2 (u(t))] for a.e. t ∈ I


w(0) = e0
Setting E(u) := w we have then defined an operator E : W 1,p (I) −→ RI called
generalized play operator. In [20, §III.2] it is proved that E is a hysteresis operator
and [20, Theorems III.2.3, III.2.4, V.2.2] show that if γi is Lipschitz continuous,
i = 1, 2, then E(W 1,p (I)) ⊆ W 1,p (I) and E is continuous with respect to the strict
metric. If the functions γi are not regular but γ1 < γ2 , by the same theorems we
have that the play operator has a continuous extension from BV (I) ∩ C(I) into
itself. In both cases it is locally monotone. Therefore Theorem 3.1 applies and this
e : BV (I) −→ BV (I). This
entails the existence of a unique continuous extension E
extension result for the play operator was obtained, with different techniques, in [6]
and [14]. However our result applies to every rate independent operator which is
locally isotone, and in fact the next applications we are going to show yield new
extension results.
5.2. The Preisach operator. We begin by describing the relay operators. Let
ρ = (ρ1 , ρ2 ) ∈ R2 be such that ρ1 < ρ2 . Let η ∈ {−1, 1}. The relay operator
Rρ,η : BV (I) ∩ C(I) −→ RI is defined by setting



−1 if u(0) ≤ ρ1
Rρ,η (u)(0) := η
if ρ1 < u(0) < ρ2


1
if u(0) ≥ ρ2
RATE INDEPENDENT OPERATORS
15
and
Rρ,η (u)(t) :=



Rρ,η (u)(0) if u(s) 6∈ {ρ1 , ρ2 } ∀s ∈ [0, t]
−1


1
if max{s ∈ [0, t] : u(s) ∈ {ρ1 , ρ2 }} = ρ1 .
if max{s ∈ [0, t] : u(s) ∈ {ρ1 , ρ2 }} = ρ2
Let P = {(ρ1 , ρ2 ) ∈ R2 : ρ1 < ρ2 } be the so-called Preisach plane and let µ be a

µ
({ρ ∈ P : ρi = r}) = 0 for every r ∈ R
finite Borel measure on P satisfying 
and for i = 1, 2. Let η : P −→ {−1, 1} a Borel measurable function. The Preisach
operator P : BV (I) ∩ C(I) −→ RI is defined by
Z
P(u)(t) :=
Rρ,η(ρ) (u)(t) dµ(ρ)
P
In [7, Proposition 2.4 and Theorem 2.5] it is proved that P maps BV (I) ∩ C(I) into
itself and that it is a hysteresis operator. Moreover [6, Proposition 2.4.14] (or [20,
Remark p. 144]) implies that P is continuous with respect to the strict metric in
BV (I). If µ is a positive measure, then P is locally monotone (cf. [7, Proposition
2.4]), hence we can apply Theorem 3.1 and extend continuously P to the space
BV (I) in a unique way.
5.3. The Duhem operator. Let w0 ∈ R and let gi : R2 −→ R be continuous,
i = 1, 2, such that there exists L ∈ C(R) positive such that |gi (r, s1 ) − gi (r, s2 )| ≤
L(r)|s1 − s2 | for every r, s1 , s2 ∈ R and i = 1, 2. For any u ∈ W 1,p (I), p ∈ [1, ∞] let
w =: D(u) be the unique solution of the Cauchy problem
(
w0 (t) = g1 (u(t), w(t))(u0 (t))+ − g2 (u(t), w(t))(u0 (t))− in I
w(0) = w0
(here x+ and x− represent the positive and the negative part of a number x ∈
R). In this way we have defined an operator D : W 1,p (I) −→ RI that is causal
and rate independent (cf. [20, Theorem V.1.1]). By [20, Theorem V.1.2] D maps
W 1,p (I) into itself and is continuous with respect to the strict metric. Finally if
g1 and g2 are positive then D is locally monotone (see [20, p. 140]). Under these
assumptions we can apply Theorem 3.1 and infer the existence of a unique continuous
extension of D to the whole space BV (I). Notice that this is not in contrast with
the counterexample in [20, p. 136], where g1 (r, s) = 2r and g2 ≡ 0 and it is shown
that D cannot be extended by continuity to BV (I): indeed g1 is not positive so that
the operator is not locally monotone.
5.4. The Bouc operator. In this section we apply our extension result to the
Bouc hysteresis operator, introduced by Bouc in [3]. Although this operator is
widely used in the engineering literature (see [10, Chapter 12]), the first rigorous
mathematical analysis is only developed in [18]. This operator is defined as follows.
Let F : [0, ∞[ −→ [0, ∞[ be Lipschitz continuous, convex, and decreasing, let
16
VINCENZO RECUPERO
g ∈ C(R), and let c be a positive constant. The Bouc operator B : W 1,p (I) −→ RI ,
p ∈ [1, ∞], is defined by
Z t Z t
0
|u (τ )|dτ g(u(s))u0 (s)ds,
u ∈ W 1,1 (I).
F
B(u)(t) := cu(t) +
0
s
In [18] we proved that the operator B maps W 1,p (I) into itself, it is continuous
for the strict topology. Moreover if g ≥ 0, then B is locally monotone. Applying
in this case the extension theorem 3.1 we infer that B admits a unique extension
e : BV (I) −→ BV (I) which is continuous for the strict topology.
B
6. Appendix
In this appendix we recall some basic facts about convergence of measures. The
aim of this section is mainly to fix notations, since the terminology is not entirely
standard. We think that things are clearer and easier to write down in a general
framework. Thus we assume that X is a separable locally compact Hausdorff topological space. The family of Borel subsets of X is denoted by B(X). Let us recall
that a finite (signed) Borel measure on X is a function µ : B(X) −→ R which is
countably additive. The total variation of a finite measure µ is the smallest positive

µ
. It turns out
measure dominating the map B 7−→ |µ(B)| and will be denoted by 












+
−
that µ is finite. Setting µ := (µ+µ)/2 and µ := (µ−µ)/2 we have defined

µ
 = µ+ + µ− . The vector
two finite positive measure such that µ = µ+ − µ− and 
space Mb (X) of finite (signed) Borel measures on X is a Banach space when endowed

µ
(X). By Cb (X), Cc (X), C0 (X) we denote the spaces of
with the norm kµk := 
continuous functions f : X −→ R which are respectively bounded, with compact
support, tending to 0 at infinity (i.e. for every ε > 0 the set {|f | ≥ ε} is compact).
All these spaces are endowed with the sup-norm, thus in the induced topology it
turns out that C0 (X) is the closure of Cc (X). In the reference [19] one can find
the previous notions and the Riesz RepresentationR Theorem, Rstating thatR the map
Φ : Mb (X) −→ C0 (X)0 defined by Φ(µ) : ϕ 7−→ X ϕ dµ := X ϕ dµ+ − X ϕ dµ− ,
is an isometric isomorphism, so that Mb (X) can be regarded as the dual of the
separable Banach space C0 (X). This motivates the former notion of convergence of
the following
Definition 6.1. Let µ, µn ∈ MRb (X) for all nR ∈ N. We say that µn weakly* converges
∗
to µ (notation: µn * µ) if X ϕ dµn → X ϕ dµ for every ϕ ∈ C0 (X). If this
N
convergence holds for all ϕ ∈ Cb (X) we say that µn narrowly converges to µ (µn *
µ).
The weak* convergence of measures is also called vague convergence and it corresponds to the weak* convergence of functionals on C0 (X). Therefore weak* sequential compactness results may be deduced from abstract results on weak* topology
RATE INDEPENDENT OPERATORS
17
(cf. [8]). On the other hand in general Cb (X) is not separable, so the following
result of Prokhorov is useful ([4, Theorem 1 and Theorem 2, Ch. IX, §5.3]).
Theorem 6.1. Let F be a bounded subset of Mb (X). If

µ
(X rKε ) < ε,
(6.1)
∀ε > 0 ∃Kε compact : sup
µ∈F
then F is sequentially relatively compact with respect to the narrow convergence.
Condition (6.1) is also necessary for sequential precompactness if F consists of positive measures.
Now we give a sufficient condition for narrow convergence of positive measures.
Proposition 6.1. Let (µn ) be a sequence of positive finite Borel measures on X
and let µ be a positive finite Borel measure on X such that µn (X) → µ(X) and
N
µ(A) ≤ lim inf n µn (A) for every open set A. Then µn * µ.
For a proof we refer to [4, Proposition 9, Ch. IX, §5] (Note that in that book X is
assumed to be a metric space, but the proof of the proposition holds in the general
case). We conclude with a proposition we use in our paper.
∗
Proposition 6.2. Let µ, µn ∈ Mb (X) for all n ∈ N be such that µn * µ and
 

N 

µn
*
µ
. Moreover if f is a Borel measurable function
kµn k → kµk. Then 






such that µ (Discont(f )) = 0, then
Z
Z
lim
f dµn =
f dµ.
n→∞
X
X
N
In particular µn * µ and µn (E) → µ(E) for every set E ∈ B(X) satisfying the

µ
(∂E) = 0.
condition 
Proof. From the lower semicontinuity of the norms with respect to the weak* con
 
µ
(A) ≤ lim inf n 
µn
(A) for every open set A ⊆ X,
vergence of µn it follows that 



N 



. Not let us prove the rest
hence Proposition 6.1 yields convergence µn * µ
of the proposition. If µ and µn are positive, the statement is proved in [4, Proposition 7, Ch. IX, §5]. In the general case from Theorem 6.1 and from the narrow
 
µn
 we infer that F = {µn }n satisfies (6.1), hence also the positive
convergence of 
+
measures µn and µ−
n satisfy such condition. Applying again Theorem 6.1 in the
reverse direction we infer that there exist two positive finite Borel measures ν + , ν −
N
N
− N
+
−
+
−
and a subsequence µn0 such that µ+
n0 * ν and µn0 * ν . Then µn0 * ν − ν , thus




∗
µn0
 = µn0 + µn0 *
µ
 = ν + + ν −.
µ = ν + − ν − . We have also 
ν + + ν − , therefore 




µ
, ν − ≤ 
µ
 and ν + (Discont(f )) = 0, ν − (Discont(f )) = 0. We are
Hence ν + ≤ 
then allowed to apply the result for positive measures and deduce that
Z
Z
Z
Z
+
−
+
f dν ,
lim
f dµn0 =
f dν − .
lim
f dµn0 =
0
0
n
X
X
n
X
X
18
VINCENZO RECUPERO
R
R
R
R
R
+
−
+
−
0
0
0
0
RWhence limn X f dµn = limn X f dµn0 − limn X f dµn0 = X f dν − X f dν =
f dµ and the proposition follows from the uniqueness of the limit.
X
Acknowledgments. I would like to thank F. Bagagiolo and G. Savaré for stimulating discussions and suggestions.
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Vincenzo Recupero, Dipartimento di Matematica, Politecnico di Torino, Corso
Duca degli Abruzzi, 24, 10129 Torino, Italy.
E-mail address: [email protected]