MAE 298: On Certain Foundational Issues in Feedback Theory

Abstract: The remarkable success of closed-loop control in mitigating the adverse effects of uncertainty on a system’s performance has undoubtedly enabled much of the technological world around us.
Invariably, feedback regulation is ubiquitous and can be found under the hood in the functioning of engines, the workings of biological organisms, interplanetary navigation, GPS tracking, robotics and
more. From its modest beginning during WWII years to the pivotal role of feedback control and Kalman filtering in the Apollo program that brought us to the moon, modern feedback theory has made great strides and is an important topic in graduate curricula. At the heart of the theory is the concept of controllability, that is, the quantitative assessment of the ability to control a dynamical system with the available control inputs. This talk will center on certain foundational questions in feedback theory that pertain to the simultaneous controllability of a collection of systems that remained unanswered until our recent work. In this, a main object of study is the Liouville (continuity) equation -- the PDE governing the evolution of the probability distribution of the state of a dynamical system. While it was widely believed that the basic question of controllability of the Liouville equation had been resolved, it escaped the community’s attention for almost two decades that early investigations on the subject came short of providing a satisfactory answer. In the talk, we revisit and address this topic and develop a theory for Collective Steering, the endeavor to shepherd an ensemble of dynamical systems between desired configurations using a common feedback law. Our investigation sheds light on a topological obstruction at the heart of the issue, that limits our ability to design continuous feedback control laws. Along the way, we touch upon an elegant geometric framework at the intersection of optimal transport, geometric hydrodynamics and quantum mechanics.