Controlling Multiparticle System on the Line, II - Periodic case

TL;DR

This paper proves that a classical periodic multiparticle system, like the periodic Toda lattice, can be globally controlled using a single force applied to one particle, despite differences from non-periodic systems.

Contribution

It establishes the global controllability of periodic multiparticle systems with generic interaction potentials using a single controlling force.

Findings

Controlled periodic systems differ from non-periodic ones.

Global controllability achieved with a single force for generic potentials.

Potential exceptions identified for nongeneric interaction potentials.

Abstract

As in arXiv: math. 0809.2365 we consider classical system of interacting particles $\mathcal{P}_1, ..., \mathcal{P}_n$ on the line with only neighboring particles involved in interaction. On the contrast to arXiv: math. 0809.2365 now {\it periodic boundary conditions} are imposed onto the system, i.e. $\mathcal{P}_1$ and $\mathcal{P}_n$ are considered neighboring. Periodic Toda lattice would be a typical example. We study possibility to control periodic multiparticle systems by means of forces applied to just few of its particles; mainly we study system controlled by single force. The free dynamics of multiparticle systems in periodic and nonperiodic case differ substantially. We see that also the controlled periodic multiparticle system does not mimic its non-periodic counterpart. Main result established is global controllability by means of single controlling force of the multiparticle system with ageneric potential of interaction. We study the nongeneric potentials for which controllability and accessibility properties may lack. Results are formulated and proven in Sections~2,3.