  Asking for the optimal protocol of an external control parameter that
minimizes the mean work required to drive a nano-scale system from one
equilibrium state to another in finite time, Schmiedl and Seifert ({\it Phys.
Rev. Lett.} {\bf 98}, 108301 (2007)) found the Euler-Lagrange equation to be a
non-local integro-differential equation of correlation functions. For two
linear examples, we show how this integro-differential equation can be solved
analytically. For non-linear physical systems we show how the optimal protocol
can be found numerically and demonstrate that there may exist several distinct
optimal protocols simultaneously, and we present optimal protocols that have
one, two, and three jumps, respectively.
