Outward pointing inverse Preisach operators

Michela Eleuteri ∗, Olaf Klein, Pavel Krejcı

Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin,

Germany

Abstract

In some recent papers, the concept of outward pointing operators has been used

as a tool within the mathematical investigation of equations involving hysteresis

operators and has been considered for the stop operator, the play operator, Prandtl-

Ishlinskiı operators and Preisach operators. Now, for inverse Preisach operators

conditions will be discussed which ensure that the outward pointing property is

satisfied. As an application of the theory, a stability result for solutions of a P.D.E.

with hysteresis appearing in the context of electromagnetic processes is derived.

Key words: Outward pointing operators, inverse Preisach operators, asymptotic

behaviour.

1 Outward pointing operators

We deal with mappings H : C(R+) → C(R+), where C(R+) denotes the space

of continuous functions u : R+ → R, R+ = [0,∞), endowed with the family

∗ Corresponding author: Tel. +49 30 203 72459; Fax: +49 30 204 4975; e-mail:

eleuteri@wias-berlin.de

Preprint submitted to Elsevier 13 July 2007

of seminorms

|u|[0,t] = max|u(s)|; 0 ≤ s ≤ t.

We start with the following definitions.

Definition. A mapping H : C(R+) → C(R+) is said to be:

(i) pointing outwards with bound h in the δ−neighbourhood of [A,B] for initial

values in [a, b], if for every t ≥ 0 and every u ∈ C(R+) such that

u(0) ∈ [a, b], u(s) ∈ (A− δ, B + δ), ∀ s ∈ [0, t] (1)

we have

(H[u](t)− h) (u(t)−B)+ ≥ 0,

(H[u](t) + h) (u(t)− A)− ≤ 0

(2)

for some given values of δ > 0, h ≥ 0, A ≤ a ≤ b ≤ B, where z+ = maxz, 0and z− = max−z, 0 for z ∈ R denote the positive and negative part of z,

respectively.

(ii) strongly pointing outwards for all bounds if and only if for all a ≤ b, all

δ > 0 and all h ≥ 0 one can find some A ≤ a and B ≥ b such that H is

pointing outwards with bound h in the δ−neighborhood of [A,B] for initial

values in [a, b].

Condition (2) can be explained as follows: if the value of u at time t exceeds the

limits [A,B] and the values of u in the past remained in a δ−neighbourhood

of [A,B], then H[u](t) “points outwards” with respect to the relative position

of u(t) and the interval [A,B], that is

u(t) > B ⇒ H[u](t) ≥ h ≥ 0

u(t) < A ⇒ H[u](t) ≤ −h ≤ 0.

(3)

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For more details on some recent papers dealing with the concept of outward

pointing operators, see for example [5], [6], [7], [1].

2 The Preisach operator

We denote by W 1,∞C (R+) the space of Lipschitz continuous functions λ : R+ →

R with compact support in R+ and introduce the set

Λ = λ ∈ W 1,∞C (R+) : |λ′(r)| ≤ 1 a.e..

The play operator ℘r[λ, u] generates for every t ≥ 0 a continuous state map-

ping Πt : Λ× C(R+) → Λ which with each (λ, u) ∈ Λ× C(R+) associates the

state λt ∈ Λ at time t. We recall that, following [4], we define the value of

the play first for piecewise monotone functions u defined on a bounded time

interval [0, T ] recurrently in each monotonicity interval, i.e. for λ ∈ Λ

λ0(r) = minu(0) + r, maxu(0)− r, λ(r)

and for each monotonicity interval [t∗, t∗] ⊂ [0, T ] of u

λt(r) = minu(t) + r, maxu(t)− r, λt∗(r).

By density, this definition is then extended to the whole space C(R+).

Now, consider two given functions g : R→ R and ψ : R+ × R→ R so that, if

we set

Ψ(r, s) =∫ s

0ψ(r, σ) dσ,

the following assumptions are satisfied:

Assumption 1 (i) the function g is continuous and non decreasing;

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(ii) the function ψ is locally integrable and non-negative in R+×R, Ψ(r, 0) = 0

for a.e. r > 0.

For u ∈ W 1,1loc (R+), λ ∈ Λ, and t ≥ 0 put

H[λ, u](t) = g(u(t)) +∫ ∞

0Ψ(r, ℘r[λ, u](t)) dr. (4)

The mapping H : Λ×W 1,1loc (R+) → C(R+) is called Preisach operator. We also

introduce the Preisach initial loading curve Φ : R→ R by the formula

Φ(v) = g(v) +∫ |v|

0Ψ(r, v − r sign(v)) dr (5)

= g(v) +

∫ v

0

∫ v−r

0ψ(r, σ) dσ dr if v ≥ 0

∫ −v

0

∫ v+r

0ψ(r, σ) dσ dr if v < 0.

Moreover let us set

A(r, u(0)) := maxu(0)− r, minu(0) + r, λ(r)

and introduce the following initial value mapping

A : R→ R (6)

u(0) 7→ g(u(0)) +∫ ∞

0ψ(r, A(r, u(0))) dr,

which with the initial input value u(0) associates the initial output value

H[λ, u](0).

Let us recall the following result (see [8], Section II.3, Theorem 3.17, see also

[2], Theorem 5.8).

Theorem 2 Let Assumption 1 be fulfilled and let λ ∈ Λ and β > 0 be given.

Then the operator β I +H[λ, ·] where I is the identity operator and H[λ, ·] is

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the Preisach operator introduced in (4) is invertible and its inverse is Lipschitz

continuous and clockwise oriented. Moreover the initial value mapping β I+A

is increasing and bi-Lipschitz.

From now on we set, for λ ∈ Λ and u ∈ C(R+)

G[λ, u](t) := (I +H[λ, ·])−1(u)(t). (7)

3 Outward pointing inverse Preisach

The main result of the section is the following.

Theorem 3 Let λ ∈ Λ and Assumption 1 be satisfied. Then G[λ, ·] is strongly

pointing outwards for all bounds.

Outline of the proof. Let us fix some a ≤ b, some δ > 0 and some h ≥ 0.

We check that G[λ, ·] satisfies (3) for suitable A and B.

First set fH := I + A , where A is the initial value mapping for the Preisach

operator (4) introduced in (6). Let us set

R0 := maxK, |f−1H (a)|, |f−1

H (b)|

where K > 0 is such that λ(r) = 0 for all r ≥ K. Now set

R := max R0, h + δ . (8)

We define

B := h + g(h) +∫ ∫

T+

ψ(r, v) dr dv

and

A := −h + g(−h)−∫ ∫

T−ψ(r, v) dr dv

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where

T+ := (r, v) : 0 ≤ r ≤ R; 0 ≤ v ≤ minh + r, R− r

T− := (r, v) : 0 ≤ r ≤ R; 0 ≥ v ≥ max−h− r,−R + r.

We will show that G[λ, ·] is pointing outwards with bound h in the δ−neighbourhood

of [A,B] for initial values in [a, b].

Consider some u ∈ C(R+) and some t ≥ 0 with

u(0) +H[λ, u](0) ∈ [a, b] (9)

and

A− δ ≤ u(s) +H[λ, u](s) ≤ B + δ ∀s ∈ [0, t]. (10)

In view of (3), we have to show that the following implications hold

u(t) +H[λ, u](t) > B ⇒ u(t) ≥ h

u(t) +H[λ, u](t) < A ⇒ u(t) ≤ −h.

(11)

First we remark that condition (9) implies that u(0) ∈ [f−1H (a), f−1

H (b)].

From (8), (9) and (10) it follows

umax := maxs∈[0,t]

u(s) ≤ R.

By virtue of Lemma 3.3. in [6], this in turn implies, for all s ∈ [0, t] that

℘r[λ, u](s) ≤ minu(s) + r,R− r. (12)

At this point, assuming u(t) < h, then by (12) we have that u(t)+H[λ, u](t) ≤B which we wanted to prove. The second implication in (11) is analogous. This

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yields the thesis. 2

4 Asymptotic behaviour

The main motivation for the result obtained in Section 3 is the study of the as-

ymptotic behaviour of the following P.D.E. containing the hysteresis operator

G given by (7) (omitting the dependence on λ)

ε utt + σut −4 (γ ut + G(u)) = f in Ω× (0, T ), (13)

where Ω is an open bounded set of R2 and u represents the unique non-zero

component of the magnetic induction B = (0, 0, u) (actually we consider for

simplicity planar waves and this leads to the scalar character of the model

equation). Moreover 4 is the Laplace operator, ε is the electric permittivity,

σ is the electric conductivity, γ is a suitable relaxation parameter and f is

given.

This model equation appears in the context of electromagnetic processes. In

particular it can be obtained by coupling the Maxwell equations, the Ohm

law and the following constitutive relation between the magnetic field and the

magnetic induction

H = G(u) + γ ut,

which is a rheological combination in series of a ferromagnetic element with

hysteresis and a conducting solenoid filled with a paramagnetic core. The

model is thermodynamically consistent because G is clockwise oriented.

It is possible to introduce a weak formulation for equation (13) (see Problem

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4.1 in [3]) which can be in turn interpreted as

εA−1utt + σA−1ut + γ ut + G(u) = A−1f

A−1u|t=0 = A−1u0, A−1ut|t=0= A−1v0,

(14)

a.e. in L2(Ω), where the operator A−1 is the inverse of the Laplace operator

−4u associated with the homogeneous Dirichlet boundary conditions.

It can be proved by means of a technique based on the contraction mapping

principle that problem (14) admits a unique solution u ∈ H1(0, T ; L2(Ω)) (see

[3], Section 4.3). Moreover by suitable approximation arguments it turns out

that actually

u ∈ H2(0, T ; L2(Ω)).

The main result of the section is the following.

Theorem 4 Let Assumption 1 hold and let G be given by (7) for some fixed

λ ∈ Λ. Consider the following assumptions on the initial data

u0 ∈ H1(Ω), v0 ∈ H1(Ω), (15)

and on the given function f

A−1f ∈ L2(0,∞; L2(Ω))

(A−1f)t ∈ L2(0,∞; L2(Ω))

|A−1f(t)|L1(Ω) ≤ k ∀ t ≥ 0.

(16)

Then problem (14) admits a unique solution

u ∈ L∞(Ω× (0,∞))× C(Ω× [0,∞)) (17)

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such that, if we set w := A−1u, we have the following regularity for the solution

wt ∈ L∞(Ω× (0,∞))× C(Ω× [0,∞)),

∇wt, ∇wtt ∈ L∞(0,∞; L2(Ω)),

∇wt, 4wt, ∇wtt, 4wtt ∈ L2((0,∞)× Ω).

(18)

Moreover we have

limt→∞

∫

Ω

(|G(u)|2 + |ut|2 + |utt|2

)dx = 0. (19)

Outline of the proof. The regularity of the solution (18) and the asymptotic

convergence result (19) can be obtained by testing the first equation of problem

(14) respectively by ut and utt in the scalar product of L2(Ω). The uniform

bound (17) for the solution u can be instead achieved using Theorem 3, i.e.

exploiting the outward pointing properties of the operator G. 2

References

[1] M. Brokate, O. Klein, P. Krejcı, Phys. B, 372, (2006), 5-8.

[2] M. Brokate, A. Visintin, J. Reine Angew. Math., 402 (1989), 1-40.

[3] M. Eleuteri, Nonlinear Analysis, Real World Applications, to appear.

[4] M.A. Krasnosel’skiı, A.V. Pokrovskiı, Springer, Berlin (1989), Russian edition:

Nauka, Moscow (1983).

[5] O. Klein, Appl. Math., vol. 49(4), pp. 309-341, 2004.

[6] O. Klein, P. Krejcı, Nonlin. Anal.: Real World. Appl., vol. 4, pp.755-785, 2003.

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[7] O. Klein, P. Krejcı, Phys. B, vol. 343, pp. 53-58, 2004.

[8] P. Krejcı, Tokyo: Gakkotosho, (1996).

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