Asymptotic Stability of Dissipated Hamilton-Poisson Systems

TL;DR

This paper investigates the asymptotic stability of dissipated Hamilton-Poisson systems, providing a tensorial dissipation form that preserves energy but alters the geometric structure, and analyzing stabilizability of equilibria.

Contribution

It introduces a tensorial dissipation form for Hamilton-Poisson systems that preserves the Hamiltonian and studies the asymptotic stabilizability of equilibria.

Findings

Dissipation preserves the Hamiltonian function.

Dissipation alters the Poisson geometry.

Conditions for asymptotic stabilizability of equilibria.

Abstract

We will further develop the study of the dissipation for a Hamilton-Poisson system introduced in \cite{2}. We will give a tensorial form of this dissipation and show that it preserves the Hamiltonian function but not the Poisson geometry of the initial Hamilton-Poisson system. We will give precise results about asymptotic stabilizability of the stable equilibria of the initial Hamilton-Poisson system.