Optimal Control and Geodesics on Quadratic Matrix Lie Groups

In this paper, we consider some matrix subgroups of the general linear group and in particular the special linear group that are defined by a quadratic matrix identity. The Lie algebras corresponding to these matrix groups include several classical semisimple matrix Lie algebras. We describe an optimal control problem on these groups which gives rise to geodesic flows with respect to a positive definite metric. These geodesic flows generalize the Euler equations and the symmetric representation of the geodesic flow for the n-dimensional rigid body (given earlier by Bloch, Crouch, Marsden and Ratiu), to these matrix groups. A discretization of these flows is given which gives a numerical algorithm for computation of the flow dynamics. Contents

• Type: Journal
• Year: 2008
• Where: FOCM
• Authors: Anthony M. Bloch, Peter E. Crouch, Jerrold E. Marsden, Amit K. Sanyal
• Keywords: FOCM 2008, Linear Group, Matrix, Matrix Groups