  Recent studies have explored finite-time dissipation-minimizing protocols for
stochastic thermodynamic systems driven arbitrarily far from equilibrium, when
granted full external control to drive the system. However, in both simulation
and experimental contexts, systems often may only be controlled with a limited
set of degrees of freedom. Here, going beyond slow- and fast-driving
approximations employed in previous studies, we obtain exact finite-time
optimal protocols for this unexplored limited-control setting. By working with
deterministic Fokker-Planck probability density time evolution, we can frame
the work-minimizing protocol problem in the standard form of an optimal control
theory problem. We demonstrate that finding the exact optimal protocol is
equivalent to solving a system of Hamiltonian partial differential equations,
which in many cases admit efficiently calculatable numerical solutions. Within
this framework, we reproduce analytical results for the optimal control of
harmonic potentials, and numerically devise novel optimal protocols for two
anharmonic examples: varying the stiffness of a quartic potential, and linearly
biasing a double-well potential. We confirm that these optimal protocols
outperform other protocols produced through previous methods, in some cases by
a substantial amount. We find that for the linearly biased double-well problem,
the mean position under the optimal protocol travels at a near-constant
velocity. Surprisingly, for a certain timescale and barrier height regime, the
optimal protocol is also non-monotonic in time.
