On BV-type hysteresis operators (Q1818015)

This article deals with some properties of causal hysteresis transducers \(w(t)= H(t_0, w_0)u(t)\) (the causality property means that \(H(t_0',w(t_0')) u_1(t)= H(t_0, w_0)u(t)\) for \(t_0'\leq t\leq T'\) and \(u_1(t)= u(t)\) (\(t_0'\leq t\leq T')\), if \([t_0', T']\subseteq [t_0, T]\)). The authors present some examples of such transducers (scalar and vector play operators, non-ideal relay, the Preisach operator), and study their BV-property (a transducer \(w(t)= H(t_0, w_0)u(t)\) has this property if the output \(x(t)= \Phi(u(t), w(t))\) is a bounded variation function for any continuous input \(u(t)\)). The property of lazyness is also discussed (this property was studied by A. Visintin; it means that, for any admissible input, the output is a function of minimal variation among all functions satisfying some natural conditions). The authors prove that the lazyness property (it is true for several simple hysteresis transducers) is not true for the multidimensional play in general case; however some weak variant holds. Further, a notion of \(\varepsilon\)-variation is introduced and applied, and the so-called \(p\)-variation for various \(p\geq 1\) is investigated. In the end of the article, the conditions under which the Preisach transducer has the BV-property are described. The last result states that each causal hysteresis transducer, which is continuous in the space of continuous functions and possesses the BV-property, is also continuous from \({\mathcal C}\) to \({\mathcal V}_p\) (the space of functions with finite \(p\)-variation) for any \(p> 1\).