Dynamics, Integrability and Control of Mechanical and Physical Systems

This project contributes to the fields of mechanics and control theory with the goals of advancing both the mathematical foundations and practical applications across multiple fields. It is expected to have direct industrial impact through innovations in the control and design of robotic systems, vehicles, and other complex mechanical systems. The research also enhances artificial intelligence by enabling data-driven learning of robotic dynamics and deepening the understanding of mathematical algorithms used in large language models. In addition, the project addresses key challenges in quantum computing by developing new approaches for designing and controlling quantum systems and tackling the problem of steering quantum states and controlling decoherence. The project involves collaborations with engineers and physicists, addressing research topics that include systems with motion constraints (such as cars and wheeled robots), systems with impacts (including legged robotic locomotion), and applications of machine learning in robotics. The analysis of coupled biological systems, such as interacting cells, is another focus. This research program also has a broad educational impact, informing teaching in dynamical systems, mechanics, robotics, control, and machine learning, with activities suitable for both graduate and undergraduate research projects. This project involves a study of the geometry, dynamics and control of mechanical systems including Hamiltonian and Lagrangian systems, integrable (solvable) systems, nonholonomic systems (systems with velocity constraints) and gradient flows. The investigator proposes to study the dynamics of various integrable Hamiltonian systems and related gradient flows with applications to the theory of total positivity and combinatorics. The first project addresses the study coupled Hamiltonian and gradient systems, both classical and quantum, optimal control with application to certain problems in artificial intelligence, hybrid dynamical systems (systems with smooth and discrete dynamics), and the machine learning of such systems. The second project involves the study of systems on networks and hypernetworks with applications to biological systems and interacting robotic systems. The planned work on when integrable systems are closely related to gradient systems is important in the study of total positivity and combinatorics and to the theory of stable polynomials. Total positivity in turn is related to problems in Markov chains and particle physics among other topics. The third project involves the study of integrable systems arising from optimal control problems. This involves investigating the relationship of optimal control to machine learning and the theory of coupled Hamiltonian and gradient systems including quantum control systems with external dissipation and decoherence. The fourth research thrust investigates related problems in mass transport, including hybrid systems arising in mechanics and control including systems with impacts as well optimization methods for learning such systems. This includes systems on graphs and hypergraphs with applications to synthetic biology and co-operating robotic systems. 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.