Invariant optimal control on the three-dimensional semi-Euclidean group: control affine and quadratic Hamilton-Poisson systems - Barrett, Dennis Ian (Rhodes University, Faculty of Science, Department of Mathematics, 2014)
In this thesis we consider invariant control systems and Hamilton-Poisson systems on the three dimensional semi-Euclidean group SE(1,1). We first classify the left-invariant control affine systems (under detached feedback equivalence). We provide a complete list of normal forms, as well as classifying conditions. As a corollary to this classification, we derive controllability criteria for control affine systems on SE(1,1). Secondly, we consider quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space se(1,1)*. These systems are classified up to an affine isomorphism. Six normal forms are identified for the homogeneous case, whereas sixteen representatives (including several infinite families) are obtained for the inhomogeneous systems. Thereafter we consider the stability and integration of the normal forms obtained. For all homogeneous systems, and a subclass of inhomogeneous systems, we perform a complete stability analysis and derive explicit expressions for all integral curves. (The extremal controls of a large class of optimal control problems on SE(1,1) are linearly related to these integral curves.) Lastly, we discuss the Riemannian and sub-Riemannian problems. The (left-invariant) Riemannian and sub-Riemannian structures on SE(1,1) are classified, up to isometric group automorphisms and scaling. Explicit expressions for the geodesics of the normalised structures are found.
Keywords: Automorphisms; Symmetry (Mathematics); Lyapunov stability; Geometry, Riemannian; Geometry, Affine; Elliptic functions