Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
Motivated by the ubiquity of control-affine systems in optimal control theory, we investigate the geometry of point-affine control systems with metric structures in dimensions two and three. We compute local isometric invariants for point-affine distributions of constant type with metric structures for systems with 2 states and 1 control and systems with 3 states and 1 control, and use Pontryagin's maximum principle to find geodesic trajectories for homogeneous examples. Even in these low dimensions, the behavior of these systems is surprisingly rich and varied.
Clelland, Jeanne N.; Moseley, Christopher G.; and Wilkens, George R., "Geometry of optimal control for control-affine systems" (2013). University Faculty Publications. 374.