Hadamard-like derivatives in Preisach modeling and control

We define a concept of derivatives of the relative boundary of the active set, on the Preisach plane, with respect to the time variable, under a time-varying input of the piecewise monotone type. These derivatives are related to the set-valued derivatives previously introduced by these authors. This concept of derivatives, which is akin to the derivatives introduced by Hadamard in the derivation of his formula on variation of a Green's function under perturbations of the domain for a boundary-value problem, allows the representation of the Preisach evolution in terms of a generalized variant of a differential equation, which in turn can be used as a basis for dynamic programming for optimal control and game theory problems for systems with hysteresis.