  We consider damped stochastic systems in a controlled (time-varying)
quadratic potential and study their transition between specified
Gibbs-equilibria states in finite time. By the second law of thermodynamics,
the minimum amount of work needed to transition from one equilibrium state to
another is the difference between the Helmholtz free energy of the two states
and can only be achieved by a reversible (infinitely slow) process. The minimal
gap between the work needed in a finite-time transition and the work during a
reversible one, turns out to equal the square of the optimal mass transport
(Wasserstein-2) distance between the two end-point distributions times the
inverse of the duration needed for the transition. This result, in fact,
relates non-equilibrium optimal control strategies (protocols) to gradient
flows of entropy functionals via and the Jordan-Kinderlehrer-Otto scheme. The
purpose of this paper is to introduce ideas and results from the emerging field
of stochastic thermodynamics in the setting of classical regulator theory, and
to draw connections and derive such fundamental relations from a control
perspective in a multivariable setting.
