Since the beginning of the 1990s hysteresis operators have been employed on a larger scale for the linearisation of hysteretic transducers. One reason for this is the increasing number of mechatronic applications which use solid-state actuators based on magnetostrictive or piezoelectric material or shape memory alloys. All of these actuator types show strong hysteretic effects. In addition to hysteresis, piezoelectric actuators show strong creep effects. Thus, the objective of this article is to enlarge the operator-based methodology of the hysteresis operators by elements that allow the description of systems with hysteresis and creep. To reach this objective, following the procedure used for hysteretic systems, creep operators are introduced to form, together with the hysteresis operators, the system operator for the simultaneous consideration of both phenomena. With regard to applications in control and measurement technology the existence, the uniqueness, the Lipschitz continuity and thus the input-output stability of its inverse operator are theoretically supported by functional analytical methods. Subsequently, the efficiency of this new concept is demonstrated in practice by a real-time inverse feedforward controller for piezoelectric actuators. Using this control concept, the tracking errors caused by hysteretic and creep effects are reduced by approximately one order of magnitude.

A well-known diffuse interface model for incompressible isothermal mixtures of two immiscible fluids consists of the Navier-Stokes system coupled with a convective Cahn-Hilliard equation. In some recent contributions the standard Cahn-Hilliard equation has been replaced by its nonlocal version. The corresponding system is physically more relevant and mathematically more challenging. Indeed, the only known results are essentially the existence of a global weak solution and the existence of a suitable notion of global attractor for the corresponding dynamical system defined without uniqueness. In fact, even in the two-dimensional case, uniqueness of weak solutions is still an open problem. Here we take a step forward in the case of regular potentials. First we prove the existence of a (unique) strong solution in two dimensions. Then we show that any weak solution regularizes in finite time uniformly with respect to bounded sets of initial data. This result allows us to deduce that the global attractor is the union of all the bounded complete trajectories which are strong solutions. We also demonstrate that each trajectory converges to a single equilibrium, provided that the potential is real analytic and the external forces vanish.

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