Applied Math Colloquium: Matthew Kvalheim
Title: Negative resistance in small-noise dynamics and obstructions to stabilization
Speaker: Matthew Kvalheim (University of Michigan)
Abstract: In this two-part talk I will first discuss the feedback stabilization problem of control theory, a problem of relevance in robotics along with virtually all fields of science and engineering. One important aspect of this problem is the search for obstructions to feedback stabilizability. Such obstructions yield practical “tests” relieving the expenditure of time and resources searching for stabilizing feedback laws that do not exist. After describing classical such tests due to Brockett (1983), Coron (1990), and others, I will introduce new tests which are more powerful and general than their predecessors. These new tests are derived from a new dynamical systems result asserting that, if two smooth vector fields on a manifold asymptotically stabilize the same compact set, then those vector fields are homotopic through nowhere-zero vector fields over a certain subset. I will also discuss the curious case of periodic orbit stabilizability which, interestingly, cannot be detected by any of these obstructions.
The second part of this talk is inspired by the fact that many biological, physical, and nanorobotic systems are well-modeled by Brownian particles subject to gradient dynamics plus noise. Important for many applications is the net steady-state particle current or “flux”—analogous to electric current—enabled by the noise and and an additional driving force, but this flux is rarely computable analytically. Motivated by this, I will describe joint work with Yuliy Baryshnikov investigating the steady-state flux of diffusion processes on compact manifolds. Such a flux is associated to each real 1-cohomology class and is equivalent to an asymptotic winding rate of trajectories. When the deterministic part of the dynamics is “gradient-like” in a certain sense, there is a new graph-theoretic formula for the small-noise asymptotics of the flux (based on an extension of Freidlin-Wentzell large deviations theory). When additionally the deterministic part is locally gradient and close to a generic global gradient, there is a natural flux for which the graph-theoretic formula becomes Morse-theoretic and admits a new description in terms of persistent homology. As an application, I will rigorously explain the paradoxical “negative resistance” phenomenon in Brownian transport discovered numerically by Cecchi and Magnasco (1996).