  It is well known that conservative mechanical systems exhibit local
oscillatory behaviours due to their elastic and gravitational potentials, which
completely characterise these periodic motions together with the inertial
properties of the system. The classification of these periodic behaviours and
their geometric characterisation are in an on-going secular debate, which
recently led to the so-called eigenmanifold theory. The eigenmanifold
characterises nonlinear oscillations as a generalisation of linear eigenspaces.
With the motivation of performing periodic tasks efficiently, we use tools
coming from this theory to construct an optimization problem aimed at inducing
desired closed-loop oscillations through a state feedback law. We solve the
constructed optimization problem via gradient-descent methods involving neural
networks. Extensive simulations show the validity of the approach.
