Left-invariant optimal control problems on the Heisenberg group and the associated Hamilton-Poisson systems: classification, stability and integration - Bartlett, Catherine Eve (Rhodes University, Faculty of Science, Department of Mathematics)
This thesis examines the left-invariant control affine systems of full rank, evolving on the three-dimensional Heisenberg group H3. Such systems are classified under state space equivalence, detached feedback equivalence and strongly detached feedback equivalence; a complete list of equivalence representatives is obtained. The equivalence of cost-extended control systems corresponding to left-invariant optimal control problems on H3 with fixed terminal time, affine dynamics, and affine quadratic cost is also considered. To left-invariant optimal control problems on H3 with quadratic cost, one may, via the Pontryagin Maximum Principle, associate a quadratic Hamilton-Poisson system on the (minus) Lie-Poisson space h3ô€€€. Homogeneous and inhomogeneous quadratic Hamilton-Poisson systems are investigated. These systems are classified up to an affine isomorphism. Furthermore, the stability nature of the equilibria of the systems are analysed and explicit expressions for all integral curves are determined.
Keywords: Heisenberg group; Lie groups; Feedback control systems; Hamiltonian mechanics; Poisson manifolds