Synchronization in the Frenkel-Kontorova Model with Application to Control of Nanoscale Friction

This paper tailors synchronization of multi-agent systems to motion control of the Frenkel-Kontorova (FK) model, a one-dimensional chain of harmonically coupled identical particles in a spatially periodic potential field. In particular, the goal is to drive all particles in the FK model to the desired trajectory by controlling only a single—boundary—particle. The proposed solution augments harmonic coupling in the FK model with dissipative inter-particle interactions, allowing all particles in the chain to synchronize to a particular reference trajectory. The boundary control represents a special case of pinning control. Moreover, as the FK model describes the frictional dynamics of a nanosheet sliding over a surface, we use its synchronization for controlling the nanoscale sliding friction. The key idea is to introduce a sliding reference trajectory that allows particles to move near synchrony. Synchronization effectively increases the system’s stiffness, so less energy ends up dissipated through inter-particle relative motion, thus reducing the frictional force. We validate the proposed solution through numerical simulations.