Global controllability of certain systems of rigid bodies under friction

We consider Lagrangian systems with bounded potential energy, with control constraints known in advance, and with friction. We present several sufficient conditions for the global controllability in the case of a cyclic coordinate assuming that the dry friction forces acting in the system satisfy the Amonton-Coulomb law. In particular, we prove the global controllability of a system of three equal masses successively connected by linear springs on a common rough line when the only controlling action is a bounded force applied to the first load. A majority of well-known results concerning controllability (in the traditional sense [1]) of dynamical systems were obtained only for the cases in which the continuity of the right-hand side of the differential equation is either assumed in advance or is ensured by part of the controlling resources. In the present paper, we give several sufficient conditions for the global controllability of systems of rigid bodies under the action of dry friction forces; the number of controls can be less than the number of degrees of freedom. The main idea of our approach consists in searching and using auxiliary “attainable” curves in the phase space such that these curves lie outside the domains of “nonsmoothness” of the system.