  In this contribution the optimal stabilization problem of periodic orbits is
studied in a symplectic geometry setting. For this, the stable manifold theory
for the point stabilization case is generalized to the case of periodic orbit
stabilization. Sufficient conditions for the existence of a \gls{nhim} of the
Hamiltonian system are derived. It is shown that the optimal control problem
has a solution if the related periodic Riccati equation has a unique periodic
solution. For the analysis of the stable and unstable manifold a coordinate
transformation is used which is moving along the orbit. As an example, an
optimal control problem is considered for a spring mass oscillator system,
which should be stabilized at a certain energy level.
