Geometric structures and applications to control theory and numerical integration.

Geometric mechanics refers to a variety of topics that lie at the intersection of differential geometry, dynamical systems, and analytical mechanics. The main idea is to identify the geometric structures underlying many classes of physical and engineering systems. These geometric structures can be useful for the qualitative study of the system and can also be used for instance in the design of control laws and geometric integrators. The property that a system can be derived from a variational principle is one of these useful structures. In this project we will use the inverse problem of the calculus of variations to find stabilizing controls for a variety of mechanical systems. One of the advantages of finding a variational structure is that we can then use energy methods to show stability, or find conditions for stability. We will also introduce more flexibility to the classical inverse problem to extend its possible applications. More precisely, for the inverse problem on a Lie algebra we will allow variable structure constants in order to have more freedom in the energy shaping step. The theory of exterior differential systems has been applied successfully to the inverse problem to identify variational cases. We will also adapt these techniques to the inverse problem for constrained systems with an eye towards the problem of Hamiltonization of nonholonomic systems. Finally we will also study geometric integrators for metriplectic and dissipative systems.

Keywords:GEOMETRIC MECHANICS

Disciplines:Calculus of variations and optimal control, optimisation, Differential geometry, Classical mechanics, Geometric aspects of physics, Motion planning and control