Homogeneous Hamiltonian Systems Research Articles

Dynamical Realizations of Homogeneous Hamiltonian Systems

In this work we consider homogeneous input–output systems defined by a single Volterra kernel, which have an internal state space realization in the form of a Hamiltonian system. Previous work by one of the authors considered the state space realization of input–output maps determined by finite Volterra series, the homogeneous case being characterized by an internal homogeneity. The present paper shows that minimal Hamiltonian realizations exist which exhibit the same polynomic structure whilst simultaneously displaying the canonical symplectic structure. The homogeneity is conveniently handled by extensive use of graded vector spaces, and their relation with nilpotent Lie algebras. The additional relation with symplectic structures is also utilized here. The work also provides a specialized version of the Darboux–Weinstein theorem, which is globally valid.