Optimal control of engineered swift equilibration of nanomechanical oscillators

Abstract Processes conveying physical information are always subject to speed limits. We explore the consequence of this in optimal control of indicators of finite-time thermodynamic transitions of nano-systems evolving according to an underdamped dynamics. Our main finding is that, in the presence of speed limits, problems like optimal engineered swift equilibration and transitions towards an out-of-equilibrium state at minimum work done to the system are well-posed. In particular, we show that there is no need for discontinuities in the internal energy at the end of the control horizon. This was a problem that raised doubts for a long time on the physical meaning of certain optimal transitions. We perform our analysis on the experimentally relevant case of nano-oscillators, and we expect our predictions on optimal protocols and the evolution of mean heat and work to be experimentally testable.