  Controlled Lagrangian and matching techniques are developed for the
stabilization of relative equilibria and equilibria of discrete mechanical
systems with symmetry as well as broken symmetry. Interesting new phenomena
arise in the controlled Lagrangian approach in the discrete context that are
not present in the continuous theory. In particular, to make the discrete
theory effective, one can make an appropriate selection of momentum levels or,
alternatively, introduce a new parameter into the controlled Lagrangian to
complete the kinetic matching procedure. Specifically, new terms in the
controlled shape equation that are necessary for potential matching in the
discrete setting are introduced. The theory is illustrated with the problem of
stabilization of the cart-pendulum system on an incline. The paper also
discusses digital and model predictive controllers.
