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We refer as ``Kramers-type'' dynamics to a class of stochastic
differential systems exhibiting a degenerate ``metriplectic'' structure.
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.
The control of entropy production by systems in this class differs from the now well-understood case of Langevin dynamics for two reasons. First, the entropy production as defined 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 in the form of a condition on the divergence. Using Pontryagin and Bismut control methods and restricting the attention to additive noise, we show that smooth control protocols attaining extremal values of the entropy production appear generically in continuos 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 extremals equations reduce at leading order to the Monge-Amp\`ere-Kantorovich optimal mass transport equations. |