The statistical physics of optimal control theory

In this introductory talk, I will present my personal perspective on how stochastic control theory is related to quantum mechanics, statistical inference, statistical physics and large deviation theory. It provides also my motivation for this conference. The talk consists of three parts. In the first part, I will review the ideas of Nelson and Guerra that relate the Schr\"odinger equation to a class of stochastic control problems. The second part revisits the path integral control theory that builds on the early work of Fleming and Mitter. These control problems are intimately related to statistical inference and statistical physics, The third part of the talk discusses an idea originally formulated by Schrödinger how control of a density naturally leads to a large deviation principle. The rate function is identified as the Kullback-Leibler divergence that also plays a central role in the path integral control theory.