  We refer as "Langevin-Kramers" dynamics to a class of stochastic differential
systems exhibiting a degenerate "metriplectic" structure. This means that the
drift field can be decomposed into a symplectic and a gradient-like component
with respect to a pseudo-metric tensor associated to random fluctuations
affecting increments of only a sub-set of the degrees of freedom. Systems in
this class are often encountered in applications as elementary models of
Hamiltonian dynamics in an heat bath eventually relaxing to a Boltzmann steady
state.
  Entropy production control in Langevin-Kramers models differs from the now
well-understood case of Langevin-Smoluchowski dynamics for two reasons. First,
the definition of entropy production stemming from fluctuation theorems
specifies a cost functional which does not act coercively on all degrees of
freedom of control protocols. Second, the presence of a symplectic structure
imposes a non-local constraint on the class of admissible controls. Using
Pontryagin control theory and restricting the attention to additive noise, we
show that smooth protocols attaining extremal values of the entropy production
appear generically in continuous parametric families as a consequence of a
trade-off between smoothness of the admissible protocols and non-coercivity of
the cost functional. Uniqueness is, however, always recovered in the
over-damped limit as extremal equations reduce at leading order to the
Monge-Amp\`ere-Kantorovich optimal mass transport equations.
