Time-energy trade-off in stochastic resetting using optimal control

Stochastic resetting is a driving mechanism that is known to minimize the first passage time to reach a target at the cost of energy expenditure. The choice of the physical implementation of each resetting event determines the trade-off between the acceleration of the search process and its energetic cost. Here we use an optimal transport protocol that balances the duration and the energetic cost of each resetting event. This protocol drives a harmonically trapped Brownian particle between two equilibrium states within a finite time and with minimal energetic cost. An explicit comparison with other types of finite-time protocols further shows its specific thermodynamic properties. Its cost is both a lower bound on the cost of unoptimized shortcut protocols and an upper bound on the cost of optimal protocols which do not ensure final equilibrium. When applying the optimal transport protocol to implement stochastic resetting, a single lower time-energy bound is reached: This protocol allows to reach the best trade-off between energetic cost and search time.