Dynamics, Integrability, and Control of Mechanical and Physical Systems

The project focuses on mechanical systems and control and has many applications in industry, such as in the control of robotic systems and design of vehicles and other complex mechanical systems, the design and control of quantum systems used in quantum computing, and the analysis of coupled biological systems such as interacting cells. The investigator will collaborate with engineers and physicists at University of Michigan.

Some of the research focuses on systems with motion constraints which includes cars and robots with wheels, while other parts focus on systems with impacts which includes robots with legs and the study of legged locomotion. There are also applications to machine learning in robotics particularly with respect to vision. In the quantum realm the investigator will study the problem of steering one quantum state to another which is key in quantum computing.

In the biological regime the investigator intends to study the control of cell type, which is key, for example, to stem cell research. The program has a strong educational impact and ideas from it will be used in teaching various subjects, including dynamical systems, mechanics, robotics and control. Parts of the proposed activity are appropriate for PhD projects and undergraduate projects.

The present project is a continuation of the investigator's study 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 with applications to certain problems in artificial intelligence, systems with impacts and nonholonomic systems, optimal control equations on manifolds in both the smooth and discrete setting, and the control and dynamics of quantum systems and certain biological systems.

He will analyze the geometry of integrable systems in various new contexts including systems arising from certain optimal control problems including flows on Stiefel manifolds, extensions of the Toda lattice flow and rigid body flows. He will also study related gradient flows which have applications to certain problems in computer vision and artificial intelligence.

In addition, he will study Hamiltonian systems with added mechanical dissipation. Further work concerns the control and optimal control of quantum systems with Lindblad dissipation which model open quantum systems and which have application to problems in quantum computing. He will also study systems with impacts (hybrid dynamical systems) including application to nonholonomic systems with impacts.

The theory of nonholonomic dynamics is the study of mechanical systems subject to constraints imposed on velocities. Such constraints often occur in robotic systems. In addition, he will consider the existence of periodic behavior in various systems including systems with impacts and certain biological systems that arise in synthetic biology.

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.