  We study the convergence to equilibrium of an underdamped Langevin equation
that is controlled by a linear feedback force. Specifically, we are interested
in sampling the possibly multimodal invariant probability distribution of a
Langevin system at small noise (or low temperature), for which the dynamics can
easily get trapped inside metastable subsets of the phase space. We follow
[Chen et al., J. Math. Phys. 56, 113302, 2015] and consider a Langevin equation
that is simulated at a high temperature, with the control playing the role of a
friction that balances the additional noise so as to restore the original
invariant measure at a lower temperature. We discuss different limits as the
temperature ratio goes to infinity and prove convergence to a limit dynamics.
It turns out that, depending on whether the lower ("target") or the higher
("simulation") temperature is fixed, the controlled dynamics converges either
to the overdamped Langevin equation or to a deterministic gradient flow. This
implies that (a) the ergodic limit and the large temperature separation limit
do not commute in general, and that (b) it is not possible to accelerate the
speed of convergence to the ergodic limit by making the temperature separation
larger and larger. We discuss the implications of these observation from the
perspective of stochastic optimisation algorithms and enhanced sampling schemes
in molecular dynamics.
