Collaborative Research: Structural Frameworks for Output Control of Continuum Ensemble Systems

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Ensemble control is about using a common control input to simultaneously steer a large population (in the limit, a continuum) of dynamical systems. These dynamical systems can themselves be networked multi-agent systems. Because the framework is by nature resilient and scalable, ensemble control finds many applications across various disciplines in science and engineering, ranging from quantum spin systems to neuroscience, emergent behavior, and multi-robot control. Many existing works in this area have focused on the problem of controlling an ensemble system as a whole. However, one is often interested in only controlling a certain output function, such as the mean. At the current stage, little is known for output control of an ensemble system. This project aims to fill the gap. The goal of the project is to establish new tool and results to usher in a framework for output control of continuum ensemble systems. The research program will integrate ideas from multiple areas in mathematics, such as control theory, graph theory, Lie theory, functional analysis, and dynamical systems. The research findings of this project will thus push forward the development of all these research areas. The focus on continuum ensembles of networked systems makes the output control framework adaptable to address problems in various application domains as mentioned above.This project will revolve around several central objects, namely, the output function associated with an ensemble system, the sparsity pattern shared by the individual systems, the geometry of the parameterization space, and the type of system dynamics. Specifically, the PIs will formulate two complementary research thrusts targeting two important classes of output functions and investigate their relations with the other objects: (1) In the first thrust, the PIs will introduce and address a new controllability notion for ensemble systems, termed sub-profile controllability. Roughly speaking, a sub-profile of an ensemble system can be understood as the collection of certain substates of the individual systems. Amongst others, the PIs will aim to establish necessary and sufficient conditions for a sub-profile to be controllable and to develop algorithms for testing. (2) In the second thrust, the PIs will study output functions that integrate individual systems of an ensemble system. Typical examples of such output functions include the average and, more generally, higher moments. The research findings of this thrust will provide insights for the problem of understanding the statistical properties of dynamic profiles over continuum parametrization space.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.