  Frictional forces are a key ingredient of any physical description of the
macroscopic world, as they account for the phenomena causing transformation of
mechanical energy into heat. They are ubiquitous in nature, and a wide range of
practical applications involve the manipulation of physical systems where
friction plays a crucial role. In this paper, we apply control theory to
dynamics governed by the paradigmatic rate- and state-variable law for
solid-on-solid friction. Several control problems are considered for the case
of a slider dragged on a surface by an elastic spring. By using swift
state-to-state protocols, we show how to drive the system between two arbitrary
stationary states characterized by different constant sliding velocities in a
given time. Remarkably, this task proves to be feasible even when specific
constraints are imposed on the dynamics, such as preventing the instantaneous
sliding velocity or the frictional force from exceeding a prescribed bound. The
derived driving protocols also allow to avoid a stick-slip instability, which
instead occurs when velocity is suddenly switched. By exploiting variational
methods, we also address the functional minimization problem of finding the
optimal protocol that connects two steady states in a specified time, while
minimizing the work done by the friction. We find that the optimal strategy can
change qualitatively depending on the time imposed for the duration of the
process. Our results mark a significant step forward in establishing a
theoretical framework for control problems in the presence of friction and
naturally pave the way for future experiments.
