# Rate Independent Evolutions

#### Abstract

An operator \( \mathcal{W} \) which maps time-dependent input functions \( u \) into time-dependent output functions \( w = \mathcal{W}[u] \) is called rate independent if it commutes with arbitrary monotone time transformations \( \varphi \) of the underlying time interval, \( W[u \circ \varphi] = (W[u]) \circ \varphi \). If, for example, \( u \) is replaced by \( \tilde u(t) = u(2t) \) , the corresponding output becomes \( \tilde w(t) = w(2t) \).
Rate independent operators can be specifed explicitly, or they arise as solution operators of dynamical systems described, for instance, by evolution variational inequalities. Examples include the relay (thermostat), the play (backlash) as well as constitutive models in continuum mechanics and electromagnetism. Usually they occur as part of a larger (not rate independent) dynamical system. This leads to coupled systems of differential equations of standard type and of rate independent elements.
The course is intended to give an introduction into the basics of rate independence and into diverse aspects of such coupled systems. No prior knowledge of rate independence is assumed. The mathematical tools used come from differential equations, functional analysis and convex analysis. In addition to the topics mentioned above, problems of optimization and control of such systems will be discussed.

#### Date and Place