Preisach models can be defined as a weighted superposition of kernels \([k_s(v,\xi](t)\) where \(v\) and \(\xi\) respectively denote inputs and initial states. The closed Preisach plane \(S=(s_1,s_2)\) is given by \[ S = \{\left(s_1,s_2 \right)| s_1 \le s_2 \} \] or the compact subset \[ S_\Delta = \{\left(s_1, s_2\right)| \underline{s}_1 \le s_1 \le s_2 \le \overline{s}_2 \}. \] Denoting the weights by \nu(s), the general can be formulated as \[ \lbrack F_\nu\left(v,\xi\right)\rbrack (t) = \int \int_S \lbrack k_s \left(v,\xi\right)\rbrack(t) \nu(s) ds \] with an analogous definition of \(S_\Delta\) when decay in \(\nu(s)\) is invoked to truncate the domain.
It is advantageous when establishing convergence properties for approximation methods or providing a framework which includes related models, such as that of Prantl, to consider the more general formulation \[ \lbrack F_\mu \left(v,\xi \right)(t) = \int \int_S \lbrack k_s \left(v, \xi(s) \right) \rbrack (t) d \mu(s), \] whre \(\mu \in \mathbb{M}\) is a signed Borel measure in the set \(\mathbb{M} \) of all finite, signed Borel measures on \(S\).
The classic Preisach model employs the multivalued relay operator \[ \lbrack k_s \left(v,\xi \right) \rbrack (t) = \begin{cases} \begin{matrix} k_s \left(v,\xi \right) \rbrack (0) & \text{ if } \tau(t) = \emptyset \\ -1 & \text{ if } \tau(t) \ne \emptyset \text{ and } v\left(\text{max }{\tau(t)}\right)=s_1 \\ +1 & \text{ if } \tau(t) \ne \emptyset \text{ and } v\left(\text{max }{\tau(t)}\right)=s_2 \end{matrix} \end{cases} \] where the crossing times \(\tau(t)\) are defined by \[ \tau (t) =\{t_s \in (0,t] | v(t_s) = s_1 \text{ or } v(t_s) = s_2 \} \] The starting value \[ \lbrack k_s \left(v,\xi \right) \rbrack (0) = \begin{cases} \begin{matrix} -1 & \text{ if } v(0) \le s_1 \\ \xi & \text{ if } s_1 < v(0) < s_2\\ +1 & \text{ if } v(0) \le s_1 \end{matrix} \end{cases} \] defines the initial states of the kernel in terms of the parameter \( \xi \in \{-1,1\}\).
The second choice of kernel illustrates the construction of a general Krasnosel'skii-Pokrovskii (K=P) operator which provides improved continuity, approximation, and convergence properties. To defined the K-P kernel, we consider translates \(r(v-s_1)\) and \(r(v-s_2)\) of a Lipschitz continuous ridge function \(r(v)\). For time interval \( \lbrack t_{k-1}, t_k \rbrack \) where the input \(v\) is monotone, a monotone operator is recursively defined by \[ \lbrack \mathcal{R}(v,\mathcal{R}_{k-1} \rbrack(t) = \begin{cases} \begin{matrix} \text{max}\{\mathcal{R}_{k-1},r\left(v(t)-s_2\right) \} \text{ if } v \text{ is non-decreasing} \\ \text{max}\{\mathcal{R}_{k-1},r\left(v(t)-s_1\right) \} \text{ if } v \text{ is non-increasing} \end{matrix} \end{cases} \] where \[ \mathcal{R}_k = \begin{cases} \begin{matrix} \mathcal{R}\left(v,\mathcal{R}_{k-1}\right) (t_k), k=2,\dots,j \\ \mathcal{R}_0= \xi, k=1, \xi \in \{-1, 1\} \end{matrix} \end{cases} \] defines the values of \(\mathcal{R}\) at times \(t_k\). The K-P is then defined recursively on each subinterval by \[ \lbrack k_s\left(v,\xi\right)\rbrack (t) = \lbrack \mathcal{R}\left(v, \mathcal{R}_{k-1}\right) \rbrack (t) \in \lbrack t_{k-1},t_k \rbrack. \]
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