Embedded geodesic problems and optimal control for matrix Lie groups

Abstract

This paper is devoted to a detailed analysis of the geodesic problem onmatrix Lie groups, with left invariant metric, by examining representationsof embeddings of geodesic flows in suitable vector spaces.We show how these representations generate extremals for optimal control problems.In particular we discuss in detail the symmetric representation of the so-called$n$-dimensional rigid body equation and its relation to the moreclassical Euler description.We detail invariant manifolds of these flows on which we are able todefine a strict notion of equivalence between representations, andidentify naturally induced symplectic structures.