# Optimal work in a harmonic trap with bounded stiffness

## TL;DR

This paper uses optimal control theory to determine the minimal work required to transition a Brownian particle between states in a harmonic trap with bounded stiffness, revealing new solution types and practical implications.

## Contribution

It extends previous models by incorporating bounded stiffness constraints, showing how they affect optimal protocols and work minimization in thermodynamic processes.

## Key findings

- Bounded stiffness can prevent certain state transitions.
- Different solution regimes depend on operation time and compression ratio.
- Work minimization is significantly impacted by stiffness bounds.

## Abstract

We apply Pontryagin's principle to drive rapidly a trapped overdamped Brownian particle in contact with a thermal bath between two equilibrium states corresponding to different trap stiffness $\kappa$. We work out the optimal time dependence $\kappa(t)$ by minimising the work performed on the particle under the non-holonomic constraint $0\leq\kappa\leq\kappa_{\max}$, an experimentally relevant situation. Several important differences arise, as compared with the case of unbounded stiffness that has been analysed in the literature. First, two arbitrary equilibrium states may not always be connected. Second, depending on the operating time $t_{\text{f}}$ and the desired compression ratio $\kappa_{\text{f}}/\kappa_{\text{\i}}$, different types of solutions emerge. Finally, the differences in the minimum value of the work brought about by the bounds may become quite large, which may have a relevant impact on the optimisation of heat engines.