Abstract A non-linear controllable dynamical system of general form, described by Lagrange's equations, is considered. The generalized control forces are subject to geometrical constraints. It is required to construct feedback-implementable control forces that will steer the system in finite time from an arbitrary initial state to a given terminal state. The problem has an explicit solution under fairly general assumptions. The construction utilizes a decomposition of the system into several simpler subsystems, each with one degree of freedom. It is shown, in particular, that if the system is subject to control forces alone, it can be steered in finite time to any given state, however weak these forces. Upper bounds are obtained for the duration of the control process.