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Generalized Finite-time Optimal Control Framework in Stochastic Thermodynamics

Optimal processes in stochastic thermodynamics are a frontier for understanding the control and design of non-equilibrium systems, with broad practical applications in biology, chemistry, and nanoscale/mesoscale systems. Optimal mass transport theory and thermodynamic geometry have emerged as optimal control methodology, but they are based on slow-driving and close to equilibrium assumptions. An optimal control framework in stochastic thermodynamics for finite time driving is still elusive. Therefore, we solve in this paper an optimal control problem for changing the control parameters of a discrete-state far-from-equilibrium process from an initial to a final value in finite-time. Optimal driving protocols are derived that minimize the total finite-time dissipation cost for the driving process. Our framework reveals that discontinuous endpoint jumps are a generic, model-independent physical mechanism that minimizes the optimal driving entropy production, whose importance is further amplified for far-from-equilibrium systems. The thermodynamic and dynamic physical interpretation and understanding of discontinuous endpoint jumps is formulated. An exact mapping between the finite-time to slow driving optimal control formulation is elucidated, developing the state-of-the-art of optimal mass transport theory and thermodynamic geometry, which has been the current paradigm for studying optimal processes in stochastic thermodynamics that relies on slow driving assumptions. Our framework opens up a plethora of applications to the thermodynamically efficient control of a far-from-equilibrium system in finite-time, which opens up a way to their efficient design principles.