Controlling Klein-Gordon Chains and Lattices

TL;DR

This work develops a finite-time control framework for nonlinear Klein–Gordon chains and lattices on a fixed discrete grid, showing that every physically admissible flock state can be reached from arbitrary initial vibrations using explicit $M$-bounded feedback controls. The authors formulate the problem on a discrete torus with a cubic nonlinearity, define moving and stationary flock states, and prove three theorems that construct explicit control laws to drive the system to desired flocks, including synchronized configurations and traveling waves with velocity $\bar v=\sqrt{\alpha/\beta}$. A Lyapunov-based strategy underpins the first theorem, guaranteeing nonincreasing energy and finite-time convergence, while the subsequent results extend the construction to staged maneuvers with robust conditions on the target flock. Finally, they connect the finite-time flocking problem to a minimal-time optimal-control framework via a high-dimensional Hamilton–Jacobi equation, offering a unifying perspective for wave control in discrete nonlinear media and laying groundwork for future sparse and infinite-dimensional extensions.

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.

Theorems & Definitions (16)

• Definition 1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Remark 1
• Remark 2
• Lemma 2.2
• proof