A Survey of the Hysteretic Duhem Model

  1. 1.Department of Mathematics, Barcelona East School of EngineeringUniversitat Politècnica de CatalunyaBarcelonaSpain
Original Paper

DOI: 10.1007/s11831-017-9218-3

Cite this article as:
Ikhouane, F. Arch Computat Methods Eng (2017). doi:10.1007/s11831-017-9218-3


The Duhem model is a simulacrum of a complex and hazy reality: hysteresis. Introduced by Pierre Duhem to provide a mathematical representation of thermodynamical irreversibility, it is used to describe hysteresis in other areas of science and engineering. Our aim is to survey the relationship between the Duhem model as a mathematical representation, and hysteresis as the object of that representation.

1 Prolegomenon

Citing a reference allows the author of a scientific article to attribute work and ideas to the correct source. Nonetheless, the process of describing that work and these ideas assumes some interpretation, at least of their relative importance. In order to ensure that the interpretation is reliable, we use, whenever adequate, a quotation from the reference so that the reader has a direct access to the cited source. This direct access is even more important when the source is not easily available like Ref. [67]1 or is not written in English like Refs. [16, 17, 18, 19, 20, 21, 22] among others, in which case we provide a translation. This is our approach to the literature review of Sect. 2.

In Sects. 49 we proceed differently since our aim in these sections is to provide an accurate description of the results presented in the references under study. Because of the diversity of notations and nomenclature in these references, quotations may not be the best way to transmit that accurate description. Instead, we summarize the references using a unifying framework provided in Ref. [35]. The references we have chosen in Sects. 49 are, in our opinion, those that are relevant to the subject of this study.

Our aim in this work is also to shed light on the relationships between the concepts introduced in this paper. To this end, we use a special form of the Duhem model, the scalar semilinear one, as a case study.

2 Introduction and Literature Review

A brief history of the Duhem model The term hysteresis was coined by J. A. Ewing in 1881 to describe a specific relationship between the torsion of a magnetized wire and its polarization (although the phenomenon of hysteresis has been known and described by several authors before that date as shown in the literature review provided in Ref. [65]).

Quoting from Ewing’s paper [28]: “These curves exhibit, in a striking manner, a persistence of previous state, such as might be caused by molecular friction. The curves for the back and forth twists are irreversible, and include a wide area between them. The change of polarization lags behind the change of torsion. To this action the author now gives the name Hysterēsis ( to be behind)”.

In 1887 Lord Rayleigh models the relationship between a magnetizing force F[Fmax,Fmax]F[Fmax,Fmax] and the corresponding magnetization M using two polynomials [59, p. 240]:
when F is decreasing,
when F is increasing, where α,α,ββ and FmaxFmax are constants.

However, the first in-depth study of hysteresis is due to Pierre Duhem2 in the period 1896–1902. A detailed review of Duhem’s work on hysteresis may be found in [67, Chapter IV] so that we provide here only those elements of that extensive work that are directly related to the present paper.

To understand the motivation for Duhem’s work we quote from [67, p. 306]: “take a metallic wire under strain by means of a load. We can take the length of the wire and its temperature as variables that define its state. The gravity weight P will represent the external action. At temperature T and under the load P the wire may be at equilibrium with length l. Give P infinitely small variations, the length l and temperature T will also experience infinitely small variations, and a new equilibrium may be achieved. In this last state, give the gravity weight and temperature variations equal in absolute value, but of opposite signs to the previous ones. The length l should experience a variation equal to the previous one with opposite sign. However, experimentation shows that this is not the case. In general, to the expansion of the wire corresponds a smaller contraction, and the difference lasts with time.”

This permanent deformation is the subject of a seven-memoirs research by Duhem, see Refs. [16]–[22]. In his first memoir submitted to the section of sciences of the Académie de Belgique on October 13, 1894, and reviewed by the mathematician Charles Lagrange in Ref. [44],3 Duhem writes: “The attempts to make the different kinds of permanent deformations compatible with the principles of thermodynamics have been few up till now. Only one of these attempts, due to M. Marcel Brillouin, appears to us worthy of interest.” [16, p. 3]. Duhem analyzes the work of Brillouin and concludes that it is not compatible with the principles of thermodynamics [16, p. 6] (see also [19, pp. 5–7]).

As an alternative, Duhem starts a theory of permanent deformations by considering the simplest case: that of a system defined by one normal variable x and its absolute temperature T. Denoting F(x,T)F(x,T) the internal thermodynamic potential of the system, Duhem writes [16, p. 8]: “Let X be the external action to which this system is subject. The condition of equilibrium of the system will be
Let (xTX) and (x+dx,T+dT,X+dX)(x+dx,T+dT,X+dX)be two equilibria of the system, infinitely close to each other; owing to equality [(1)] we get
Equation (2) does not take into account the fact that the modifications of equilibria are not reversible. So Duhem introduces a term f(xTX)|dx| to be added to the right-hand side of Eq. (2), where f is a continuous function of the three variables x, T, and X. For an isothermal modification (that is when T is maintained constant) we get [16, pp. 9–10]:
dXdx={f1(x,T,X) for an increasing x,f2(x,T,X) for a decreasing x,
dXdx={f1(x,T,X)f2(x,T,X) for an increasing x, for a decreasing x,
Observe that, when the input is piecewise monotone, the model (3) is equivalent to the model (5) proposed in Refs. [3] and [43, p. 282]:
˙x(t)={ϕ(x(t),u(t))˙u(t) for ˙u(t)0,ϕr(x(t),u(t))˙u(t) for ˙u(t)0,
x˙(t)={ϕ(x(t),u(t))u˙(t)ϕr(x(t),u(t))u˙(t) for u˙(t)0, for u˙(t)0,
where ϕϕ and ϕrϕr are functions that satisfy some conditions, the function u is the input (which is x using Duhem’s notation), the function x the state (which is X using Duhem’s notation), and t is time.

To the best of our knowledge, the first reference that called the form (5) “Duhem model” is Ref. [48] in 1993.4 Indeed, the authors of Ref. [43] attributed erroneously Duhem’s model to Madelung [63, p. 797].5

Between 1916 when P. Duhem dies and 1993 when his model of hysteresis is finally attributed to him, Duhem’s work on hysteresis does not have a relevant impact. Major references on hysteresis like Refs. [8, 12] or [56] do not cite his memoirs. Several authors propose different forms of the Duhem model without a direct reference to Duhem’s memoirs. This is the case of the Coleman and Hodgdon model of magnetic hysteresis [12], the Dahl model of friction [13], the model (5) in Ref. [3], and a generalized form of the model (5) in Ref. [43, p. 95]. In 1952, Everett cites briefly Duhem’s work as follows [24, p. 751]: “From a thermodynamic standpoint the introduction of an additional variable whose value depends on the history of the system is sufficient for a formal discussion (cf. Duhem [ref][ref]). To advance our understanding of the phenomenon [of hysteresis], however, a molecular interpretation is desirable.”

A general theory of physics based on a molecular interpretation was precisely what Duhem rejected. In a review of his work presented in 1913 for his application to the Académie des Sciences, Duhem writes that his “doctrine should note imitate the numerous mechanical theories proposed by physicists hitherto; to the observable properties that apparatus measure, it will not substitute hidden movements of hypothetical bodies” [67, p. 74].

In recent times, Duhem’s phenomenological approach is becoming more accepted [5, 9, 46, 52, 57]. Indeed, “hysteretic phenomena arising in structural and mechanical systems are so complicated that there has been no well-accepted mathematical model which can describe all observed hysteretic characteristics.” [52, p. 1408]. Moreover, the Preisach model which was believed to describe the constitutive behavior of magnetic hysteresis, has shown to be a phenomenological model [49, p. 2].

Several reasons are invoked for the use of Duhem’s model to describe hysteresis. On the one hand, “differential equation-based models lead to a particularly simple phenomenological description” [46, p. C8–545]. On the other hand, the “Duhem models [sic] have the advantage that [they] require a small amount of memory so they are suitable in practical and low cost applications.” [9, p. 628]. Finally, many phenomenological models of friction or hysteresis can be seen as particular cases of a more general form of the Duhem model: this is the case for example of the Dahl [13], the LuGre [2, 11], or the Maxwell-slip models [30]. Thus “recast[ing] each model in the form of a generalized Duhem model provide[s] a unified framework for comparing the hysteretic nature of these models.”s [57, p. 91].

There are several generalizations of the original Duhem model (5). The following generalization is proposed in [43, p. 95]: ˙x(t)=f(t,x(t),u(t),˙u(t))x˙(t)=f(t,x(t),u(t),u˙(t)). In [64, p. 141] the terms ϕ(x(t),u(t))ϕ(x(t),u(t)) and ϕr(x(t),u(t))ϕr(x(t),u(t)) in (5) are replaced by [ϕ(x,u)](t)[ϕ(x,u)](t) and [ϕr(x,u)](t)[ϕr(x,u)](t) respectively, where ϕϕ and ϕrϕr are causal operators. In Ref. [54] Duhem’s model is generalized as ˙x(t)=f(x(t),u(t))g(˙u(t))x˙(t)=f(x(t),u(t))g(u˙(t)) whilst [64, p. 145] proposes the following form for vector hysteresis: ˙x(t)=f(x(t),u(t),π(˙u))|˙u(t)|x˙(t)=f(x(t),u(t),π(u˙))|u˙(t)| where π(˙u0)=˙u/|˙u|π(u˙0)=u˙/|u˙|.

Why are there different generalized forms of the Duhem model ? To answer this question we have to recall the concept of rate independence.6

To the best of our knowledge, the earliest author to state clearly rate independence is R. Bouc in Ref. [8], although that property was known before Bouc’s work. Due to the importance of rate independence in the study of hysteresis, and the fact that Ref. [8] is not available in English, we quote from [8, p. 17]:“Consider the graph with hysteresis of Fig. 1 where FFis not a function of x. To the valuex=x0x=x0correspond four values ofFF.
Fig. 1

Graph “Force–displacement” with hysteresis

If we consider now x as a function of time, the value of the force at instant t will depend not only on the value x(t), but also on all past values of function x since the origin instant where it is defined. If ββ is that instant (x(β)=F(β)=0,βx(β)=F(β)=0,β), then we denote F(t)=A(x(),t)F(t)=A(x(),t) the value of the force at instant “tt”, where x()x() “represents” the whole function on the interval [β,t][footnote][β,t][footnote]. Our aim is to explicit functional A(x(),t)A(x(),t).

To this end, we make the following assumption: the graph of Fig. 1 remains the same for all increasing function x()x() between 0 and x1x1, decreasing between the values x1x1 and x2x2, etc. The functional will no longer depend explicitly on time and we will write F(t)=A(x())(t)F(t)=A(x())(t). We can say: If x(tj)x(tj) and x(tj+1)x(tj+1) are two extremal values, consecutive in time, we have for all t[tj,tj+1]t[tj,tj+1]
where fjfj is a function of only the variable x(t).
We can also say: If ϕ:RRϕ:RR is a class C1C1 function whose derivative is strictly positive for tβtβ with ϕ(β)=βϕ(β)=β, and if we consider the function y(t)=x(ϕ(t))y(t)=x(ϕ(t)) which is a “compression” or an “expansion” of x by intervals, then the graphs (A(y()),y)(A(y()),y) and (A(x()),x)(A(x()),x) are identical and we have
The exact definition of rate independence varies from author to author. For example, Visintin requires the time-scale-change ϕϕ to be a strictly increasing time homeomorphism [64, p. 13] whilst Oh and Bernstein consider that ϕϕ is continuous, piecewise C1C1, nondecreasing, ϕ(0)=0ϕ(0)=0, and limtϕ(t)=limtϕ(t)= [54]. Loosely speaking, rate independence means that the graph of hysteresis (output versus input) is invariant with respect to any change in time scale.

Rate independence is used by Visintin to define hysteresis :“Definition. Hysteresis = Rate Independent Memory Effect.”[64, p. 13]. However, “this definition excludes any viscous-type memory” [64, p. 13] because it leads to rate-dependent effects that increase with velocity. A definition based on rate independence assumes that “the presence of hysteresis loops is not … an essential feature of hysteresis.” [64, p. 14].

This point of view is challenged by Oh and Bernstein who consider hysteresis as a “nontrivial quasi-dc input-output closed curve” [54, p. 631] and propose a modified version of the Duhem model which can represent rate-dependent or rate-independent effects. A characterization of hysteresis systems using hysteresis loops is also addressed by Ikhouane in Ref. [35] through the concepts of consistency and strong consistency.

In light of what has been said it becomes clear that, in Ref. [64], the generalizations of Duhem’s model are done in such a way that rate independence is preserved, whilst a definition of hysteresis based on hysteresis loops in Ref. [54] is compatible with a generalized form of the Duhem model that may be rate dependent or rate independent.

Why are there different models of hysteresis? In Ref. [16] Duhem proposes his model to account for the irreversibility in the modifications of equilibria observed experimentally in magnetic hysteresis [16, Chap. IV], sulfur [17], red phosphorus [19, Chap. III], and in different processes of metallurgy [19].

Preisach [56] uses “plausible hypotheses concerning the physical mechanisms of magnetization” [49, p. 1] to elaborate a model of magnetic hysteresis. This model is also proposed and studied by Everett and co-workers [24, 25, 26, 27] who postulate “that hysteresis is to be attributed in general to the existence in a system of a very large number of independent domains, at least some of which can exhibit metastability.” [24, p. 753].

Krasnosel’skiǐ and Pokrovskiǐ point out to the issue of admissible inputs, as “it is by no means clear a priori for any concrete transducer with hysteresis, how to choose the relevant classes of admissible inputs” [43, p. 5]. This is why they introduce the concept of vibro-correctness which allows the determination of the output of a hysteresis transducer that corresponds to any continuous input, once we know the outputs that correspond to piecewise monotone continuous inputs [43, p. 6]. The models that Krasnosel’skiǐ and Pokrovskiǐ propose (ordinary play, generalized play, hysteron) are vibro-correct, although the authors acknowledge the existence of hysteresis models that may not be vibro-correct like the Duhem model.7

Hysteresis models based on a feedback interconnection between a linear system and a static nonlinearity are proposed in Ref. [55]. The authors study “hysteresis arising from a continuum of equilibria … and hysteresis arising from isolated equilibria” [55, p 101].

A review of hysteresis models is provided in Ref. [48] and a detailed study of these (and other) models may be found in Refs. [7, 10, 14, 37], [49, 64].

In light of what has been said, the diversity of hysteresis models is due to the wide range of areas to which hysteresis is concomitant, and the diversity of methods and assumptions underlying the elaboration of these models.

Note that all mathematical models of hysteresis share a common property: they model hysteresis. This fact leads us to our next question.

What is hysteresis? A description found in many papers is that hysteresis “refers to the systems that have memory, where the effects of input to the system are experienced with a certain delay in time.” [33, p. 210]. This description is misleading as it applies also to dynamic linear systems. Indeed, when the output y is related to the input u by ˙x=Ax+Bux˙=Ax+Bu and y=Cxy=Cx which is a possible description of a linear system, the output is given by y(t)=C[exp(tA)x0+t0exp((tτ)A)Bu(τ)dτ]y(t)=C[exp(tA)x0+t0exp((tτ)A)Bu(τ)dτ] where x0x0 is the initial state and t0t0 is time. We can see that y(t) depends on the integral of a function that incorporates u(τ)u(τ) for all τ[0,t]τ[0,t], which means that the linear system does have memory. However, “hysteresis is a genuinely nonlinear phenomenon” [10, p. 7].

Mayergoyz considers hysteresis as a rate-independent phenomenon which is “consistent with existing experimental facts.” [49, p. 16]. However, “for very fast input variations, time effects become important and the given definition of rate-independent hysteresis fails.” [49, p. 16]. Also, “in the existing literature, hysteresis phenomenon is by and large linked with the formation of hysteresis loops (looping). This may be misleading and create the impression that looping is the essence of hysteresis. In this respect, the given definition of hysteresis emphasizes the fact that history dependent branching constitutes the essence of hysteresis, while looping is a particular case of branching.” [49, pp. 16–18].

Following Mayergoyz, “All rate-independent hysteresis nonlinearities fall into two general classifications: (a) hysteresis nonlinearities with local memories, and (b) hysteresis nonlinearities with nonlocal memories.” [49, pp. 17]. In a hysteresis with a local memory, the state or output at time tt0tt0 is completely defined by the state or output at instant t0t0, and the input on [t0,t][t0,t]. This is the case for example of a hysteresis given by a differential equation. Hysteresis with a nonlocal memory is a hysteresis which is not with local memory. This is the case for example of the Preisach model. “However, the notion of hysteresis nonlinearities with local memories is not consistent with experimental facts.” [49, pp. 19–20]. Hodgdon, on the other hand, writes in relation with the use of a special case of the Duhem model to represent ferromagnetic hysteresis: “These results are in good agreement with the manufacturer’s dc hysteresis data and with experiments” [34, p. 220].

In Ref. [54], Oh and Bernstein consider the generalized Duhem model ˙x=f(x,u)g(˙u)x˙=f(x,u)g(u˙) and y=h(x,u)y=h(x,u) with u the input, y the output and x the state. The authors assume the existence of a unique solution of the differential equation on the time interval [0,[[0,[. They also assume the existence of a T–periodic solution xTxT for any T–periodic input uTuT with one increasing part and one decreasing part, which means that the graph {(uT,xT)}{(uT,xT)} is a closed curve. Finally they assume that when TT the graph {(uT,xT)}{(uT,xT)} converges with respect to the Hausdorff metric to a closed curve CC. If we can find (a,b1)C(a,b1)C and (a,b2)C(a,b2)C with b1b2b1b2, the curve CC is not trivial and the generalized Duhem model is a hysteresis.

In a PhD thesis advised by Bernstein [15], Drinčić considers systems of the form ˙x=f(x,u)x˙=f(x,u) and y=h(x,u)y=h(x,u) for which hysteresis is defined as in Ref. [54]. The system is supposed to be step convergent, that is limtx(t)limtx(t) exists for all initial conditions and for all constant inputs. It is noted that there exists “a close relationship” [15, p. 6] between the curve CC and the input-output equilibria map, that is the set E={(u,h(limtx(t),u))}E={(u,h(limtx(t),u))} where u is constant and f(limtx(t),u)=0f(limtx(t),u)=0. In particular, the “system is hysteretic if the multivalued mapEEhas either a continuum of equilibria or a bifurcation” [15, p. 7].

In Ref. [6] Bernstein states that “a hysteretic system must be multistable; conversely, a multistable system is hysteretic if increasing and decreasing input signals cause the state to be attracted to different equilibria that give rise to different outputs.” Multistability means that “the system must have multiple attracting equilibria for a constant input value” [6].

In Ref. [50], Morris presents six examples of hysteresis systems taken from the areas of electronics, biology, mechanics, and magnetics; hysteresis being understood as a “characteristic looping behavior of the input-output graph” [50, p. 1]. The author explains the qualitative behavior of these systems from the point of view of multistability. For “the differential equations used to model the Schmitt trigger, cellular signaling and a beam in a magnetic field” it is observed that “these systems, all possess, for a range of constant inputs, several stable equilibrium points.” [50, p. 13]. The author observes that the systems are rate dependent for high input rates.

For the play operator, the Preisach model and the Bouc-Wen model which are rate independent, “these models present a continuum of equilibrium points.” [50, p. 13]. These observations lead the author to conclude that “hysteresis is a phenomenon displayed by forced dynamical systems that have several equilibrium points; along with a time scale for the dynamics that is considerably faster than the time scale on which inputs vary.” [50, p. 13]. Morris proposes the following definition.

“A hysteretic system is one which has (1) multiple stable equilibrium points and (2) dynamics that are considerably faster than the time scale at which inputs are varied.” [50, p. 13].

In Ref. [35], Ikhouane considers a hysteresis operator “HHthat associates to an input u and initial conditionξ0ξ0 an output y=H(u,ξ0)y=H(u,ξ0), all belonging to some appropriate spaces.” [35, p. 293]. It is assumed that the operator HH is causal and satisfies the property that constant inputs lead to constant outputs. Examples include all rate-independent models [47, Proposition 2.1], some rate-dependent models, models with local memory like the various generalizations of the Duhem model, and models with nonlocal memory like the Preisach model.

The author introduces two changes in time scale: (1) a linear one which is applied to a given input, and (2) a -possibly- nonlinear one which is the total variation of the original input. When the input is composed with the linear time-scale change, both the input and the output are re-scaled with respect to the total variation of the input, which provides a normalized input independent of the linear time-scale change, and a normalized output. Consistency is defined as being the convergence of the normalized outputs in the space LL endowed with the uniform convergence norm. It is shown that consistency implies the convergence to some set of the graphs output versus input of the hysteresis operator when the linear time scale varies [35, Lemma 9].

Strong consistency is defined as the property that the limit of the normalized outputs, seen a parametrized curve, converges to a periodic orbit which characterizes the hysteresis loop.

The author does not propose a definition of hysteresis, but considers that consistency and strong consistency are properties of a class of hysteresis systems.

Aim of the paper The aim of the paper is to survey the research carried out on the Duhem model from the perspective of its hysteretic properties.

Organization of the paper Section 4 presents some results obtained in Ref. [43], namely the concept of vibro-correctness, sufficient conditions to ensure global solutions of the scalar rate independent Duhem model, and a study of the continuity of the model seen as an operator. Section 5 presents a definition of hysteresis proposed in Ref. [54] that uses a generalized form of Duhem’s model as a tool to get that formal definition. Section 6 presents the concepts of consistency and strong consistency introduced in Ref. [35]. The tools and notations of Ref. [35] are also used as a unifying framework to present the results of the present paper. Section 7 presents a characterization of the generalized Duhem model obtained in Ref. [51]. Section 8 summarizes the results obtained in Ref. [40] in relation with the study of the dissipativity of the Duhem model. Section 9 summarizes some results obtained in Ref. [64] in relation with the existence of a Duhem operator, its smoothness, and some generalizations of the model. Section 10 is a note that explores the minor loops of hysteresis systems with particular emphasis on the Duhem model. For ease of reference, some results on the existence and uniqueness of the solutions of differential equations are presented in Appendix 15.

To illustrate the results obtained in Sects. 410, and to analyze the relationships between these results, we use the scalar semilinear Duhem model as a case study. The corresponding mathematical analysis is stated in various lemmas and theorems provided in Section 11, whose proofs are given in 1620. The relationships between the results obtained in Sects. 49 are commented upon in Section 12. These comments lead to the formulation of several open problems in Sect. 13 and a conjecture in Sect. 11.9.

3 Terminology and Notations

A real number x is said to be strictly positive when x>0x>0, strictly negative when x<0x<0, nonpositive when x0x0, and nonnegative when x0x0. A function h:RRh:RR is said to be strictly increasing when t1<t2h(t1)<h(t2)t1<t2h(t1)<h(t2), strictly decreasing when t1<t2h(t1)>h(t2)t1<t2h(t1)>h(t2), nonincreasing when t1<t2h(t1)h(t2)t1<t2h(t1)h(t2), and nondecreasing when t1<t2h(t1)h(t2)t1<t2h(t1)h(t2).8

An ordered pair ab is denoted (ab) whilst the open interval {tRa<t<b}{tRa<t<b} is denoted ]ab[. The set of nonnegative integers is denoted N={0,1,}N={0,1,} and the set of nonnegative real numbers is denoted R+=[0,[R+=[0,[.

The Lebesgue measure on RR is denoted μμ. We say that a subset of RR is measurable when it is Lebesgue measurable. Let IR+IR+ be an interval, and consider a function ϕ:IRlϕ:IRl where l>0l>0 is an integer. We say that ϕϕ is measurable when ϕϕ is (Mμ,B)(Mμ,B)–measurable where B is the class of Borel sets of RlRl and MμMμ is the class of measurable sets of R+R+ [66]. For a measurable function ϕ:IRlϕ:IRl, ϕIϕI denotes the essential supremum of the function |ϕ||ϕ| on I where |||| is the Euclidean norm on RlRl. When I=R+I=R+, this essential supremum is denoted ϕϕ.

W1,(R+,Rl)W1,(R+,Rl) denotes the Sobolev space of absolutely continuous functions ϕ:R+Rlϕ:R+Rl. For this class of functions, we have ϕ<ϕ<; the derivative of ϕϕ is denoted ˙ϕϕ˙; this derivative is defined almost everywhere and satisfies ˙ϕ<ϕ˙<. Endowed with the norm ϕW1,(R+,Rl)=max(ϕ,˙ϕ)ϕW1,(R+,Rl)=max(ϕ,ϕ˙), the vector space W1,(R+,Rl)W1,(R+,Rl) is a Banach space [45, pp. 280–281].

L(R+,Rl)L(R+,Rl) denotes the Banach space of measurable and essentially bounded functions ϕ:R+Rlϕ:R+Rl endowed with the norm .

C0(R+,Rl)C0(R+,Rl) denotes the Banach space of continuous functions ϕ:R+Rlϕ:R+Rl endowed with the norm .

γ]0,[γ]0,[, the linear time-scale-change sγ:R+R+sγ:R+R+ is defined by the relation sγ(t)=t/γ,tR+sγ(t)=t/γ,tR+.

limxalimxa sets for limxax<alimxax<a whilst limxalimxa sets for limxax>alimxax>a.

Let U be a set and let T]0,[T]0,[. The function ϕ:R+Uϕ:R+U is said to be T–periodic if ϕ(t)=ϕ(t+T),tR+ϕ(t)=ϕ(t+T),tR+.

4 A Summary of the Results Obtained in Ref. [43]

This section presents those results obtained in Ref. [43] that are relevant to the present paper. This is in particular the case of the concept of vibro-correctness which allows to extend the set of admissible inputs from continuously differentiable to continuous.

4.1 The Concept of Vibro-Correctness

Consider the differential equation [43, p. 95]
In Eqs. (6)–(7) the initial time t0Rt0R and the initial state x0Rnx0Rn where n>0n>0 is an integer. Furthermore, the function ζ1:R×Rn×R×RRnζ1:R×Rn×R×RRn is continuous and the input uC1([t0,[,R)uC1([t0,[,R). Theorem 10 ensures the existence of at least a solution of (6)–(7) on some time interval [t0,t0[[t0,t0[ where t0>t0t0>t0 may be finite or infinite. Is it possible to extend the set of inputs from continuously differentiable to solely continuous? The answer to this question leads to the concept of vibro-correctness.
Let t1]t0,[t1]t0,[ and vC0([t0,t1],R)vC0([t0,t1],R). For any δ]0,[δ]0,[ define the set

Definition 1

[43, pp. 95–96] The differential equation (6)–(7) is vibro-correct if for each x0Rnx0Rn and each input uC0([t0,[,R)uC0([t0,[,R) there exist t1]t0,[t1]t0,[ and δ0]0,[δ0]0,[ such that Propreties (i)–(ii) hold.
  1. (i)

    uE(δ0,u)uE(δ0,u) the solution x=W(u,x0)x=W(u,x0) of Eqs. (6)–(7) exists and is unique on the time interval [t0,t1][t0,t1].

  2. (ii)



In the following we analyze the consequences of vibro-correctness. Consider a sequence of inputs ukE(δ0,u)ukE(δ0,u) such that limkuku[t0,t1]=0limkuku[t0,t1]=0. Then, owing to Proprety (ii) of Definition 1, it follows that {W(uk,x0)}kN{W(uk,x0)}kN is a Cauchy sequence in C0([t0,t1],R)C0([t0,t1],R). Thus it converges with respect to the norm to a function xC0([t0,t1],R)xC0([t0,t1],R). Note that the function xx is independent of the particular choice of the sequence ukuk owing to Proprety (ii) of Definition 1. Defining W(u,x0)W(u,x0) as being xx means that the operator WW has been extended to the set of continuous inputs.

Thus, the concept of vibro-correctness allows to extend the definition of the operator WW from the set of continuously differentiable inputs to that of continuous inputs.

Another consequence of Property (ii) is the uniqueness of the solutions of (6)–(7). This means that it is not necessary to state explicitly in Property (i) that the differential equation (6)–(7) has a unique solution (this is what is done in Ref. [43]; see also [43, p. 104]).

Definition 2

[43, p. 98] If we consider only constant inputs uu in Definition 1 then the differential equation (6)–(7) is said to be vibro-correct on constant inputs.

Theorem 1

[43, p. 98] If the differential equation (6)–(7) is vibro-correct on constant inputs then we can find functionsζ2,ζ3:R×Rn×RRnζ2,ζ3:R×Rn×RRn such that for all(t,x,u,v)R×Rn×R×R(t,x,u,v)R×Rn×R×R we haveζ1(t,x,u,v)=ζ2(t,x,u)v+ζ3(t,x,u)ζ1(t,x,u,v)=ζ2(t,x,u)v+ζ3(t,x,u).

Theorem 1 means that the only differential equations (6)–(7) that may be vibro-correct are the ones that have the following form:

4.2 Global Solutions of the Scalar Rate-Independent Duhem Model

Consider the space S(t0,t2)S(t0,t2) of absolutely continuous functions u:[t0,t2]Ru:[t0,t2]R such that
where t2]t0,[t2]t0,[ is fixed. Consider following differential equation [43, p. 286]:
˙x(t)=h(x(t),u(t))˙u(t) for almost all t[t0,t2] such that ˙u(t)0,
x˙(t)=h(x(t),u(t))u˙(t) for almost all t[t0,t2] such that u˙(t)0,
˙x(t)=hr(x(t),u(t))˙u(t) for almost all t[t0,t2] such that ˙u(t)0,
x˙(t)=hr(x(t),u(t))u˙(t) for almost all t[t0,t2] such that u˙(t)0,
where uS(t0,t2)uS(t0,t2), and x(t)Rx(t)R. The functions h,hr:R×RRh,hr:R×RR are Borel, locally bounded,9 and satisfy the following unilateral Lipschitz conditions with respect to the first variable [43, p. 278]:
where λ:RR+λ:RR+ is continuous, au=mint[t0,t2]u(t)au=mint[t0,t2]u(t), and bu=maxt[t0,t2]u(t)bu=maxt[t0,t2]u(t). Observe that (15)–(16) are the transcription of (133) for the differential equation (12)–(13). Given that the function λλ is continuous, it is bounded on the interval [au,bu][au,bu] so that the term λ(v)λ(v) in Inequalities (15)–(16) can be replaced by a constant. Thus there exists a unique solution to (12)–(14) whose maximal interval of existence is [t0,t2][t0,t2] owing to Lemma 12.

4.3 Continuity of the Rate-Independent Duhem Model Seen as an Operator

For any given initial condition x0Rx0R define the operator Zx0:S(t0,t2)S(t0,t2)Zx0:S(t0,t2)S(t0,t2) that associates to each input uS(t0,t2)uS(t0,t2) the solution x of the differential equation (12)–(14). Then,

Theorem 2

[43, Theorem 29.1] The operatorZx0Zx0 is continuous. Furthermore, leta]0,[a]0,[, then


5 A Summary of the Results Obtained in Ref. [54]

This section presents those results obtained in Ref. [54] that are relevant to the present paper. In particular, the authors of Ref. [54] propose a definition that decides whether a given generalized Duhem model is a hysteresis or not.

5.1 The Generalized Duhem Model

The generalized Duhem model with input u, state x and output y consists of a differential equation that describes the state x as [54]
˙x(t)=f(x(t),u(t))g(˙u(t)), for almost all tR+,
x˙(t)=f(x(t),u(t))g(u˙(t)), for almost all tR+,
and an algebraic equation that describes the output y as
In Eqs. (17)–(19) the input uW1,(R+,R)uW1,(R+,R);10 the function f:Rn×RRn×nf:Rn×RRn×n is continuous; n and nn are strictly positive integers; the function g:RRng:RRn is continuous and satisfies g(0)=0g(0)=0; the function h:Rn×RRh:Rn×RR is continuous; and the initial state x0Rnx0Rn. The following is assumed in [54, Section II, p. 633].

Assumption 1

For every (u,x0)W1,(R+,R)×Rn(u,x0)W1,(R+,R)×Rn there exists a unique solution xW1,(R+,Rn)xW1,(R+,Rn) that satisfies Eqs. (17)–(18).

From Assumption 1 we get yC0(R+,R)L(R+,R)yC0(R+,R)L(R+,R).

Define the operator Ho:W1,(R+,R)×RnC0(R+,R)L(R+,R)Ho:W1,(R+,R)×RnC0(R+,R)L(R+,R) by the relation Ho(u,x0)=yHo(u,x