Controlling General Polynomial Networks

TL;DR

This paper establishes conditions for controllability and uniqueness of steady states in networks of particles with polynomial interactions, extending previous results to more complex network structures and providing constructive criteria based on topology and force inequivalence.

Contribution

It provides new sufficient, constructive conditions for controllability and steady state uniqueness in polynomial particle networks, generalizing earlier simpler chain results.

Findings

Conditions for Hormander's bracket condition are derived.

Controllability and steady state uniqueness are established.

Examples include conducting chains with variable cross-section.

Abstract

We consider networks of massive particles connected by non-linear springs. Some particles interact with heat baths at different temperatures, which are modeled as stochastic driving forces. The structure of the network is arbitrary, but the motion of each particle is 1D. For polynomial interactions, we give sufficient conditions for H\"ormander's "bracket condition" to hold, which implies the uniqueness of the steady state (if it exists), as well as the controllability of the associated system in control theory. These conditions are constructive; they are formulated in terms of inequivalence of the forces (modulo translations) and/or conditions on the topology of the connections. We illustrate our results with examples, including "conducting chains" of variable cross-section. This then extends the results for a simple chain obtained in Eckmann, Pillet, Rey-Bellet (1999).