Derivatives of rate-independent evolutions

There are three main approaches to rate-independent evolutions: hysteresis operators (initiated by M.A. Krasnoselskii), sweeping processes (J.-J. Moreau) and energetic systems (A. Mielke). A basic setting common to all three approaches is given by the so-called stop operator and play operator, which are nonsmooth operators between appropriate function spaces. We discuss the existence and form of derivatives (mainly, directional derivatives) of the stop and the play, and relate them to optimization problems.