Title: Universal solution for optimal protocols of finite-time and weak processes

Abstract: The analytical expression for the optimal protocol of the thermodynamic work and its variance for finite-time, isothermal and weak processes is presented. The method that solves the Euler-Lagrange integral equation is quite general and depends only on the properties of the time-reversal symmetry of the optimal protocol. The solution is proven to be physically consistent and many examples are solved to illustrate the method. To overcome the hypothesis consistency problem of the singular part of the solution, an interpretation of the appearance of the delta peaks and their derivatives is presented.