Control of friction: shortcuts and optimization for the rate- and state-variable equation

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.