Realization Theory for Hamiltonian Systems

Abstract

We prove that control systems of the Hamiltonian type have a quasi-minimal realization on a state space of minimal dimension, of one of two types: a Hamiltonian system or the suspension of a time-dependent Hamiltonian system. These realizations are unique up to a symplectomorphism, in the first case, or a canonical transformation, in the second one. When the Lie algebra of the initial system $\Sigma $, assumed to be analytic, is finite dimensional, we obtain a characterization of the state space and the dynamics of the quasi-minimal realization in terms of the coadjoint actions of Lie groups associated with $\Sigma $; an example is given applying these results. Using control theoretic methods, we prove the Kostant–Kirillov–Souriau theorem; the technique used for systems with finite dimensional Lie algebra is closely related to the theory developed by those authors.