  We reconsider a control theory for Hamiltonian systems, that was introduced
on the basis of KAM theory and applied to a model of magnetic field in previous
articles. By a combination of Frequency Analysis and of a rigorous (Computer
Assisted) KAM algorithm we prove that in the phase space of the magnetic field,
due to the control term, a set of invariant tori appear, and it acts as a
transport barrier. Our analysis, which is common (but often also limited) to
Celestial Mechanics, is based on a normal form approach; it is also quite
general and can be applied to quasi-integrable Hamiltonian systems satisfying a
few additional mild assumptions. As a novelty with respect to the works that in
the last two decades applied Computer Assisted Proofs into the framework of KAM
theory, we provide all the codes allowing to produce our results. They are
collected in a software package that is publicly available from the {\it
Mendeley Data} repository. All these codes are designed in such a way to be
easy-to-use, also for what concerns eventual adaptations for applications to
similar problems.
