Optimal constrained control for generally damped Brownian heat engines

Optimization of cyclic stochastic heat engines, a topic spanning decades of research, commonly assumes fixed control or response parameters at discrete points in the cycle-a limitation that often leads to experimentally impractical protocols. We overcome this with a general algorithm, adapted from optimal control theory, that optimizes full-cycle dynamics under realistic constraints, such as stiffness and temperature bounds, across diverse systems. Unlike geometric or mass transport methods, which rely on fixed endpoints and are unsuitable for unconstrained cycles, our approach simultaneously tunes both cycle time and control variations. Applied to a generally damped Brownian particle in a harmonic potential-an experimentally relevant case-our method is validated in the overdamped regime and extended to arbitrary damping rates. As damping decreases, maximum power vanishes and cycle time diverges; at fixed cycle times, efficiency follows a similar trend, with optimal protocols exhibiting non-monotonic complexity. Notably, optimizing temperature profiles-often overlooked-significantly enhances efficiency in intermediate damping regimes. Our work establishes the first systematic framework for optimizing cyclic stochastic processes under experimental constraints, broadening the scope of power and efficiency optimization in nonequilibrium thermodynamics.