Dynamics, Integrability, and Control of Mechanical and Nonholonomic Systems

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The project focuses on problems in mechanics, dynamics, and control. The theory of nonholonomic dynamics is the study of mechanical systems subject to constraints on velocities, such as rolling without slipping. Examples where we use this theory in practical systems include wheeled vehicles, such as cars (in particular self-steering cars) and robots. The mathematics behind the control of nonholonomic systems plays a key role in control of mechanical systems in general, such as the control of aircraft. Also important is how dissipation, or friction, affects the behavior and stability of such systems. This research project explores how to explicitly solve for the dynamics to predict the behavior of such mechanical systems. The methods under development are expected also to be useful in studying quantum control problems, with applications to quantum computing among others. The project involves graduate and undergraduate students in the research. This research project aims to broaden and deepen understanding of the geometry, dynamics, and control of mechanical systems including Hamiltonian and Lagrangian systems, integrable systems, nonholonomic systems, and gradient flows. The investigator will study the dynamics of various mechanical systems including integrable Hamiltonian systems in finite and infinite dimensions, coupled Hamiltonian and gradient systems, systems with nonholonomic constraints, optimal control equations on manifolds, and quantum control systems. The research will consider the geometry of integrable systems in several new contexts, including extensions of the Toda lattice flow and rigid body flows, as well as applications to optimal control of certain systems on Lie groups. It is expected that similar methods can be used to study the control and dynamics of open quantum systems that involve coupled Hamiltonian and dissipative dynamics.