**2. Hysteresis operators: Basic properties**

In this section, we summarize the basic knowledge of scalar hysteresis operators based on the description given in [6].

Most systems experiencing hysteresis phenomena display *hysteresis loops*. To illustrate this in a simple setting, let us consider a system like the one depicted in **Figure 1** (left), whose state is characterized by two time-dependent scalar variables, *u* and *w*. Let us also suppose that *u* is the independent variable, and thus the evolution of *w* depends on *u*. Such diagrams implicitly show the *memory effect* inherent to the hysteresis: at any time *t*>0, *w t*ð Þ depends on the evolution of *u* in 0, ½ �*t* , rather than only on *u t*ð Þ. In many cases, not all the information in 0, ½ �*t* is used to compute the value of *w* but only the one which has not been *wiped-out* from the memory (see Section 3.1). In other words, the state of the system at present time depends on the previous *history* of its state. Consequently, the same instantaneous values of the input can give different outputs and when describing the relation between *w* and *u* in the ð Þ *u*, *w* plane a multi-branch nonlinearity appears with branch to branch transitions (see **Figure 1**, left).

According to Ref. [6], two main characteristics of hysteresis phenomena can be distinguished: *causality* and *rate independence*. To clarify these concepts, we consider the ð Þ *u*, *w* relation introduced above. *Causality* means that, at any time *t*, the value of *w t*ð Þ only depends on the previous evolution of *u*, namely, *w t*ð Þ depends on *u*j½ � 0,*<sup>t</sup>* . On the other hand, *rate independence* is translated by the condition that, at any instant *t*, *w t*ð Þ depends just on the range of function *u* : ½ �! 0, *t* and on the order in which the values of *u* have been attained. In other words, *w* is independent of the *velocity* of *u*. An example is given in **Figure 1** (right) where it is shown that three different functions *u* : ½ �! 0, *t* lead to the same *w* � *u* curve.

**Figure 1.** *Hysteresis loop (left) and rate independence example (right).*

We notice that, even in the most typical hysteresis phenomena, like plasticity, ferroelectricity or ferromagnetism, the memory effect is not completely rateindependent, since viscous-type effects are coupled with hysteresis. However, in the applications presented in the following sections, this effect is neglected, so that the rate-independent component prevails.

To provide a functional setting for hysteresis operators, we first notice that, at any instant *t*, *w t*ð Þ will depend not only on the previous evolution of *u* (i.e., on *u*j½ � 0,*<sup>t</sup>* ), but also on the "initial state" of the system. Due to the memory dependence of hysteresis processes, additional information is needed to make up for the lack of history when the process begins. This initial information must represent the "history" of the function *u* before *t* ¼ 0. Hence, not only the standard initial value ð Þ *u*ð Þ 0 , *w*ð Þ 0 must be provided. In general, a variable *ξ* containing all the information about the "initial state" is considered. This can be expressed, for instance, as follows:

$$
\tilde{\varphi}(u,\xi) \to w = \tilde{\mathcal{F}}(u,\xi). \tag{1}
$$

In the case of partial differential equations, it is necessary to define an operator F acting between suitable function spaces involving the space variable. Let Ω^ be an open subset of *<sup>N</sup>*ð Þ *<sup>N</sup>* <sup>≥</sup><sup>1</sup> ; given a hysteresis operator <sup>F</sup><sup>~</sup> , we introduce, for any *<sup>u</sup>*: <sup>Ω</sup>^ � ½ �! 0, *<sup>T</sup>* and any *<sup>ξ</sup>* : <sup>Ω</sup>^ ! *<sup>Y</sup>*, with *<sup>Y</sup>* a suitable space, the corresponding space dependent operator F as follows (see [23] for a rigorous mathematical setting)

$$\mathbb{E}\left[\mathcal{F}(u,\xi)\right](\mathbf{x},t) \coloneqq \left[\tilde{\mathcal{F}}(u(\mathbf{x},\cdot),\xi(\mathbf{x}))\right](t), \qquad \forall t \in [0,T], \quad \text{a.e. in } \hat{\Omega}. \tag{2}$$

We notice that operator <sup>F</sup><sup>~</sup> is local in space, i.e., the output ½ � <sup>F</sup>ð Þ *<sup>u</sup>*, *<sup>ξ</sup>* ð Þ *<sup>x</sup>*, *<sup>t</sup>* depends on *u x*ð Þj , � ½ � 0,*<sup>t</sup>* , but not on *u y*ð Þj , � ½ � 0,*<sup>t</sup>* for *<sup>y</sup>* 6¼ *<sup>x</sup>*.

The general setting that we have discussed so far, and the majority of the hysteresis models proposed in the literature, are scalar. Thus, they can be applied only to model unidirectional inputs. However, in many applications, the hysteresis is characterized by a vector input *u*ð Þ*t* and vector output Fð Þ *u*, *t* ; thus, vector hysteresis is encountered. This is the case, for instance, of electric devices such as actuators, transformers, or rotating machines in which the direction of the magnetic field is apriori unknown. The properties of vector hysteresis are often very different from the properties of scalar hysteresis, and the derivation of a general model of vector hysteresis remains an open question. Useful references on vector hysteresis models and their features are given in [12, 26–30].
