This talk is a guest lecture within the special topic Math 577 course on "Stochastic Control and Learning." by M. Chertkov

Abstract:         Optimal feedback control of densities, or distributions in general, is of growing interest across many science and engineering applications. The underlying mathematics has two different control-theoretic interpretations: Active control of stochastic uncertainties for a single dynamical system and shaping the population density of an ensemble of systems. The former finds application in using feedback to allow the system to meet desired statistical accuracy. The latter finds application in shaping the concentration of the dynamical agents such as in swarm control, and from this perspective, can be viewed as the continuum limit of the decentralized stochastic optimal control. In this talk, we will give an exposition of the main ideas, and clarify connections with adjacent areas such as optimal mass transport. We will outline how these ideas are becoming popular in ML algorithms.

We will then summarize the rapid developments emerging from the control literature and focus on the theory and algorithms for the case when the underlying trajectory-level evolution has prior nonlinear dynamics. It will be shown that several cases of interest to the control community, such as gradient, mixed conservative-dissipative, and feedback linearizable nonlinearities, lead to tractable theory and algorithms.  We will discuss application case studies from different areas: automated driving and self-assembly for advanced manufacturing, to illustrate the scope of the recent progress, and to highlight the exciting opportunities ahead.