PIMS Seminar: Lie group and homogeneous variational integrators and their applications to geometric optimal control theory

The geometric approach to mechanics serves as the theoretical underpinning of innovative control methodologies in geometric control theory.  These techniques allow the attitude of satellites to be controlled using changes in its shape, as opposed to chemical propulsion, and are the basis for understanding the ability of a falling cat to always land on its feet, even when released in an inverted orientation.  We will discuss the application of Lie group variational integrators to the optimal control of mechanical systems.  These methods are based on a discretization of Hamilton's principle that preserves the Lie group structure of the configuration space, and yield minimum-dimensional global representations of the dynamics.  Recent extensions to homogeneous spaces yield intrinsic methods for Hamiltonian flows on the sphere, and have potential applications to the simulation of geometrically exact rods, structures and mechanisms.  Extensions to Hamiltonian PDEs and uncertainty propagation on Lie groups using noncommutative harmonic analysis techniques will also be discussed.  In particular, this allows one to develop attitude state estimation methods for satellite dynamics without assuming that the measurements are frequent, and that the uncertainty distribution is localized.