Lyapunov stability and precise control of the frictional dynamics of a one-dimensional particle array

Using Lyapunov theory and numerical simulations, we analyze the local stability of an array of mechanically coupled particles whose frictional dynamics is described by the Frenkel-Kontorova model, and design feedback controls to precisely control the friction. We first establish the asymptotic stability of the system around the equilibrium positions of the particles. We then show how to construct efficient feedback control laws to achieve any predestined average velocity of the particle array, with no fluctuation, and irrespective of the detailed nature of the interparticle coupling. These rigorous results are supported in extensive numerical simulations, and are expected to be applicable to other related physical systems as well.