Controlling Klein-Gordon Chains and Lattices

TL;DR

This paper introduces a control framework for nonlinear Klein-Gordon chains, enabling the achievement of flocking states and coherent wave motion through feedback mechanisms, linking wave control to optimal control theory.

Contribution

It presents a novel control approach for Klein-Gordon chains that achieves flocking states and connects wave control with nonlinear Hamilton-Jacobi equations.

Findings

Any flock state can be reached in finite time.

Control mechanisms induce coherent traveling waves.

Deep connection established with optimal control theory.

Abstract

In this work, we initiate the study of controlling nonlinear Klein-Gordon chains and lattices through their emergent collective flocking behavior. By constructing appropriate feedback control mechanisms, we demonstrate that any physically admissible flock state can be achieved in finite time, meaning the chain can be driven from arbitrary initial vibrations toward a coherent traveling-wave motion. Finally, we reveal a deep connection between the flocking problem and a minimal-time control principle formulated within the framework of nonlinear Hamilton-Jacobi equations and optimal control theory, providing a unifying view-point for wave control in discrete nonlinear media.