Theory and Techniques for Controlling the Collective Behavior of Dynamical Systems under Stochastic Uncertainty

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Physical systems such as micro-mechanical components experiencing thermal noise, particle beams, and laser-driven molecular reactions can all be modeled as dynamical system with stochastic excitation. The control of such processes requires that we steer the collective behavior of the individual constituents towards a specified terminal goal. Due to the stochastic uncertainty of their initial state as well as due to various stochastic effects (measurement noise, stochastic forcing, fluctuation-dissipation phenomena, etc.) one can only hope to be able to specify a target distribution for corresponding parameters and not their precise final values. Further, for such inherently stochastic systems, it is often the case that the performance of a control strategy can only be quantified in a statistical sense. Thus, the proposed research aims at developing theory and techniques that will allow steering such systems towards a desired end-point distribution for their parameters (position, velocity, temperature, etc.) while utilizing minimal amount of resources, typically energy. A wide range of physical systems and applications are envisioned and, accordingly, a range of mathematical models with linear, nonlinear, mean-field, etc. dynamics will be studied. The tools are expected to impact technology that relates to the aforementioned physical systems and applications. A specific technical goal of the proposed research is to study the controllability and the optimal control of stochastic systems in the sense of being able to steer their state from a given initial probability density to a final target one. In effect, this is the problem to control the Fokker-Planck equation that specifies the flow of the state-probability density. Problems of optimal transport for mass and resources that obey nontrivial dynamics will also be considered; in these the stochastic uncertainty for the state remains while the stochastic excitation becomes negligible. Furthermore, problems of interacting particles and mean-field potentials will also be studied as will be the effect of observation noise and ensuing limits to performance. The development of theory and tools will make use of recent advances at the interface between Monge-Kantorovich optimal mass transport, Schroedinger bridges, and the contractive properties of stochastic maps in the geometry induced by the so-called Hilbert metric.