Phase Space Geometry of Critical Transitions in Collective Behavior Modeled by Mean Field Type Control Problems

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This grant will support research on understanding and manipulating the collective behavior of engineered large-population multi-agent systems, promoting both the progress of science and advancing national prosperity. Examples of such systems include mobile robot swarms, smart grid, metamaterial structures, and vehicular traffic. A key challenge is to impart systems such as robot swarms with the capability to autonomously switch between different collective behaviors, e.g., in response to external stimuli. In other systems such as mixed autonomous-manual traffic, it is desirable to nudge or incentivize the agents toward more desirable collective state(s), e.g., to reduce congestion. Qualitative changes in spatiotemporal collective behavior of a system are studied under the umbrella of phase transitions in physics. This research will extend such methods to controlled large-population multi-agent systems, and create a unifying framework for understanding and triggering phase transitions in such systems. This research aims to understand bifurcations and global phase space structure of non-standard dynamical systems originating in the mean field games and mean field control framework. The mean field control theory for large-population multi-agent systems combines ideas from statistical physics with optimal control, and models scenarios where a large number of interacting agents are acting optimally, either in cooperative or non-cooperative setting. The resulting dynamical systems consist of fully-coupled forward-backward in time nonlinear partial differential equations, and their complexity has to date prevented qualitative understanding of the nature of solutions. Phase transitions in the controlled collective behavior are the result of bifurcations of the solutions of closed-loop mean field control problems as problem parameters, such as cost functions, penalties, and supervisory control action, etc., are varied. This research will adapt local bifurcation theory of existing descriptive (forward) models such as the nonlinear Schrodinger equations and flocking to characterize bifurcations in prescriptive or closed-loop (forward-backward) models of mean field games and mean field control theory. The research will also produce low-order models of these infinite dimensional systems. The phase space geometry of such models will enable discovery of global bifurcations and connecting orbits responsible for switching between different collective behaviors. The use of phase space geometry to understand and induce criticality is a stepping stone towards a ‘theory of thermodynamics’ of large-scale controlled dynamical systems. The insight gained from this project can prove useful in rational design of control penalties and/or incentives to shape the collective behavior of nonlinear agents for diverse applications.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.