5 research outputs found
Geometry of Control-Affine Systems
Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-affine distribution on a manifold X – i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X) = n, rank(F) = n–1, and when dim(X) = 3, rank(F) = 1. Unlike linear distributions, which are characterized by integer-valued invariants – namely, the rank and growth vector – when dim(X) ≤ 4, we find local invariants depending on arbitrary functions even for rank 1 point-affine distributions on manifolds of dimension 2
Jacobi-Lie systems: Fundamentals and low-dimensional classification
A Lie system is a system of differential equations describing the integral
curves of a -dependent vector field taking values in a finite-dimensional
real Lie algebra of vector fields, a Vessiot-Guldberg Lie algebra. We define
and analyze Lie systems possessing a Vessiot-Guldberg Lie algebra of
Hamiltonian vector fields relative to a Jacobi manifold, the hereafter called
Jacobi-Lie systems. We classify Jacobi-Lie systems on and
. Our results shall be illustrated through examples of physical
and mathematical interest.Comment: 15 pages. Examples, references and comments added. Based on the
contribution presented at "The 10th AIMS Conference on Dynamical Systems,
Differential Equations and Applications", July 07-11, 2014, Madrid, Spain. To
appear in the Proceedings of the 10th AIMS Conferenc