  Optimal control of levitated particles subject to thermal fluctuations is a
challenging problem, both from the theoretical and the experimental
perspective. In this paper we compute the time-dependent harmonic confining
potential that steers a Brownian particle between two assigned equilibrium
states, in a prescribed time, with the minimum energetic cost. In our analysis
we do not neglect inertial effects (thus addressing the general underdamped
dynamics) and we assume, motivated by the experiments, that the stiffness of
the confining potential cannot exceed prescribed bounds. We report the results
of an experiment realizing the described protocol for an optically confined
nanoparticle. The system is shown to reach the target state within accuracy,
while spending less energy than other protocols that require the same time. The
optimal protocol is also compared to an abrupt change of the confining
stiffness, the latter involving an exponential relaxation to the target
equilibrium state. Not only does our optimal protocol reach the target state in
a finite time but it does so in a time significantly shorter than the
characteristic timescale of the direct relaxation. The results presented in
this paper are expected to have relevant applications in the design of optimal
devices, e.g. engines at the nanoscale.
