Asymptotic Stability for a Class of Metriplectic Systems

Using the framework of metriplectic systems on ${\mathbb{R}}^n$ 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.