Asymptotic Stability for a Class of Metriplectic Systems

TL;DR

This paper presents a geometric method to add dissipation to Hamilton-Poisson systems within the metriplectic framework, ensuring solutions near stable equilibria converge to an invariant set.

Contribution

It introduces a constructive approach to incorporate dissipation based solely on Hamiltonian and Casimir functions, enhancing stability analysis.

Findings

Dissipation term guarantees convergence to invariant set near stable equilibria.

Method relies only on Hamiltonian and Casimir functions.

Provides a geometric framework for stability in metriplectic systems.

Abstract

Using the framework of metriplectic systems on $\R^n$ we will describe a constructive geometric method to add a dissipation term to a Hamilton-Poisson system such that any solution starting in a neighborhood of a nonlinear stable equilibrium converges towards a certain invariant set. The dissipation term depends only on the Hamiltonian function and the Casimir functions.