Existence and optimal control of mechanical processes with hysteresis in viscous solids

In this paper, mechanical processes in a viscous one-dimensional solid are considered where a stress-strain relation in the form of a hysteresis loop is assumed. For a certain class of functionals modelling such loops, existence and stability results are established, and the existence of optimal controls for a related control problem is proved. The class of hysteresis functionals considered includes among others the hysteron of the first kind in the sense of Krasnoselskii and the Ishlinskii and Preisach operators.