  We present a stylized model of controlled equilibration of a small system in
a fluctuating environment. We derive the equations governing the optimal
control steering \emph{in finite time} the system between two equilibrium
states. The corresponding thermodynamic transition is optimal in the sense that
occurs at minimum entropy if the set of admissible controls is restricted by
certain bounds on the time derivatives of the protocols. We apply our equations
to the engineered equilibration of an optical trap considered in a recent proof
of principle experiment. We also analyze an elementary model of nucleation
previously considered by Landauer to discuss the thermodynamic cost of one bit
of information erasure. We expect our model to be a useful benchmark for
experiment design as it exhibits the same integrability properties of well
known models of optimal mass transport by a compressible velocity field.
