The optimal control of finite-time processes on the microscale is of significant theoretical and practical interest, particularly for the energy-efficient operation of nanomachines. While previous studies have primarily focused on transitions between equilibrium states, many biologically and technologically relevant processes occur far from equilibrium. In such nonequilibrium settings, memory, a ubiquitous feature in realistic systems, plays an intricate role, as any driving necessarily excites internal memory modes. This motivates a deeper exploration of optimal control strategies in nonequilibrium regimes. Here, we combine experiments, theory, and computational methods to investigate the transition of a colloidal particle confined in an optical trap between two nonequilibrium steady states (NESS). We identify optimal control protocols that minimize the thermodynamic work during the finite-time transition between two NESS. We compare optimal protocols in viscous and viscoelastic fluid environments, which are common in realistic technical and biological processes and introduce memory due to a delayed response. Regardless of the presence of memory effects, optimal protocols consistently balance energy extraction with dissipation minimization. In the presence of memory, optimal control is achieved if the protocol matches the time response of the environment. These findings offer key insights for designing optimal control strategies for finite-time, nonequilibrium processes in complex environments.
Monter et al. (Mon,) studied this question.