The geometry of Lie algebroids and its applications to optimal control

Abstract

The paper presents the geometry of Lie algebroids and its applications to optimal control. The first part deals with the theory of Lie algebroids, connections on Lie algebroids and dynamical systems defined on Lie algebroids (mainly Lagrangian and Hamiltonian systems). In the second part we use the framework of Lie algebroids in the study of distributional systems (drift less control affine systems) with holonomic or nonholonomic distributions.

Vertical and complete lifts of sections of a (dual) vector bundle and Legendre duality

“…In many papers such as [5,11,14,15], the authors studied the lifts to the second order tangent bundle, tensor bundle and jet bundle. The Lie algebroids are important issues in physics and mechanics since the extension of Lagrangian and Hamiltonian systems to their entity [6,7,12] and catching the Poisson structure [13]. Several authors presented and studied the lift of geometrical objects of a Lie algebroid ((F, ν, N ), [, ] F , (ρ, Id N )) to the Lie algebroid prolongation.…”

Models of Finsler Geometry on Lie algebroids

“…Recently, Lie algebroids are important issues in physics and mechanics since the extension of Lagrangian and Hamiltonian systems to their entity [4,17,18,20,37] and catching the poisson structure [24]. They are wrestled with nonsmooth optimization [25] and studied on Banach vector bundles [2]. They have such a flexibility that holonomy of orbit foliation carried on them [14].…”

“…The aim of this paper is rebuild the Finsler geometry concepts on Lie algebroid structures. For instance, this matter is discussed in [36,25] of course. Finsler geometry is a generalization of Riemannian geometry such that interfering of direction and position duplicates the degree of freedom in view of configuration.…”