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The complete physical understanding of the optimization of the thermodynamicwork still is an important open problem in stochastic thermodynamics. Weaddress this issue using the Hamiltonian approach of linear response theory infinite time and weak processes. We derive the Euler-Lagrange equationassociated and discuss its main features, illustrating them using theparadigmatic example of driven Brownian motion in overdamped regime. We showthat the optimal protocols obtained either coincide, in the appropriate limit,with the exact solutions by stochastic thermodynamics or can be even identicalto them, presenting the well-known jumps. However, our approach reveals thatjumps at the extremities of the process are a good optimization strategy in theregime of fast but weak processes for any driven system. Additionally, we showthat fast-but-weak optimal protocols are time-reversal symmetric, a propertythat has until now remained hidden in the exact solutions far from equilibrium.

Optimal finite-time processes in weakly driven overdamped Brownian motion