Optimal control of levitated nanoparticles subjected to thermal fluctuations is a challenging problem, both theoretically and experimentally. In this Letter, we compute the time-dependent harmonic confining potential that steers, in a prescribed time and with the minimum energetic cost, a Brownian particle between two assigned equilibrium states. We take full account of inertial effects, thus addressing the general underdamped dynamics, and, to address actual experimental conditions, the stiffness of the confining potential is required to be bounded. We carry out an experiment realizing the described protocol for an optically confined nanoparticle, which is shown to reach the target state within accuracy—while spending less energy than other protocols with the same duration, significantly shorter than the characteristic relaxation time. The results presented here are expected to have relevant applications in the design of optimal devices, such as engines at the nanoscale.

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• Brownian motion
• Optimization problems
• Stochastic thermodynamics
• Nanoparticles