  Controlling thermodynamic cycles to minimize the dissipated heat is a
longstanding goal in thermodynamics, and more recently, a central challenge in
stochastic thermodynamics for nanoscale systems. Here, we introduce a
theoretical and computational framework for optimizing nonequilibrium control
protocols that can transform a system between two distributions in a minimally
dissipative fashion. These protocols optimally transport a system along paths
through the space of probability distributions that minimize the dissipative
cost of a transformation. Furthermore, we show that the thermodynamic metric --
determined via a linear response approach -- can be directly derived from the
same objective function that is optimized in the optimal transport problem,
thus providing a unified perspective on thermodynamic geometries. We
investigate this unified geometric framework in two model systems and observe
that our procedure for optimizing control protocols is robust beyond linear
response.
