A link between contractive and dissipative Hamiltonian systems

Mario Spirito (1) , Bernhard Maschke (1) , Yann Le Gorrec (2)

1 LAGEPP - Laboratoire d'automatique, de génie des procédés et de génie pharmaceutique
2 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)

This paper should be intended as a preliminary work in which we describe a connection between contractive systems and dissipative Hamiltonian dynamics. In particular, we show that it is always possible to obtain from the Jacobian of the autonomous system, given the contraction metric, the structural matrices of a Hamiltonian dynamics. We show how to obtain in closed-form such matrices for (asymptotically) stable linear systems, by exploiting a modification of the Jordan block form. We then conclude the paper by analytically performing the analysis on an example of nonlinear contractive system.

Mots clés: Contractive systems; dissipative Hamiltonian dynamics; variational dynamics; modified Jordan form; structural matrices

Domaines: Systèmes dynamiques [math.DS]

Mario Spirito, Bernhard Maschke, Yann Le Gorrec. A link between contractive and dissipative Hamiltonian systems. 2024. ⟨hal-04496810⟩