  We consider the problem of driving a finite-state classical system from some
initial distribution $p$ to some final distribution $p'$ with vanishing entropy
production (EP), under the constraint that the driving protocols can only use
some limited set of energy functions $\mathcal{E}$. Assuming no other
constraints on the driving protocol, we derive a simple condition that
guarantees that such a transformation can be carried out, which is stated in
terms of the smallest probabilities in $\{p,p'\}$ and a graph-theoretic
property defined in terms of $\mathcal{E}$. Our results imply that a
surprisingly small amount of control over the energy function is sufficient (in
particular, any transformation $p\to p'$ can be carried out as soon as one can
control some one-dimensional parameter of the energy function, e.g., the energy
of a single state). We also derive a lower bound on the EP under more general
constraints on the transition rates, which is formulated in terms of a convex
optimization problem.
