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Controlling the Rates of a Chain of Harmonic Oscillators with a Point Langevin Thermostat

Amirali Hannani, Minh-Binh Tran, Minh Nhat Phung, Emmanuel Trélat

TL;DR

This work studies the controllability of a kinetic transport problem for an infinite chain of coupled harmonic oscillators driven by a Langevin thermostat at the origin. By analyzing two open-loop boundary controls—impulsive and linear memory-feedback—the authors show that impulsive control cannot alter the boundary reflection/transmission/absorption rates in the kinetic limit, while a memory-feedback design can explicitly modulate these rates. They derive explicit rate-function formulas $r_a^{F}$, $r_t^{F}$, $r_r^{F}$ and prove that any admissible target triple is asymptotically reachable via an appropriate control $F$, with a constructive path to approximate controls $F_N$. This provides a rigorous controllability framework for boundary-driven kinetic equations and suggests practical avenues for steering heat transport in networked oscillator systems. The results have potential implications for manipulating energy exchange and phonon transport in nanoscale lattices and related stochastic transport models.

Abstract

We consider the control problem of controlling the rates of an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. We study the effect of two types of open-loop boundary controls, impulsive control and linear memory-feedback control, in the high frequency limit. We investigate their action on the reflection-transmission coefficients for the wave energy for the scattering of the thermostat. Our study shows that the impulsive boundary controls have no impact on the rates and are thus not appropriate to act on the system, despite their physical meaning and relevance. In contrast, the second kind of control that we propose, which is less standard and uses the past of the state solution of the system, is adequate and relevant. We prove that any triple of rates satisfying appropriate assumptions is asymptotically reachable thanks to the linear memory-feedback controls that we design explicitly.

Controlling the Rates of a Chain of Harmonic Oscillators with a Point Langevin Thermostat

TL;DR

This work studies the controllability of a kinetic transport problem for an infinite chain of coupled harmonic oscillators driven by a Langevin thermostat at the origin. By analyzing two open-loop boundary controls—impulsive and linear memory-feedback—the authors show that impulsive control cannot alter the boundary reflection/transmission/absorption rates in the kinetic limit, while a memory-feedback design can explicitly modulate these rates. They derive explicit rate-function formulas , , and prove that any admissible target triple is asymptotically reachable via an appropriate control , with a constructive path to approximate controls . This provides a rigorous controllability framework for boundary-driven kinetic equations and suggests practical avenues for steering heat transport in networked oscillator systems. The results have potential implications for manipulating energy exchange and phonon transport in nanoscale lattices and related stochastic transport models.

Abstract

We consider the control problem of controlling the rates of an infinite chain of coupled harmonic oscillators with a Langevin thermostat at the origin. We study the effect of two types of open-loop boundary controls, impulsive control and linear memory-feedback control, in the high frequency limit. We investigate their action on the reflection-transmission coefficients for the wave energy for the scattering of the thermostat. Our study shows that the impulsive boundary controls have no impact on the rates and are thus not appropriate to act on the system, despite their physical meaning and relevance. In contrast, the second kind of control that we propose, which is less standard and uses the past of the state solution of the system, is adequate and relevant. We prove that any triple of rates satisfying appropriate assumptions is asymptotically reachable thanks to the linear memory-feedback controls that we design explicitly.
Paper Structure (43 sections, 12 theorems, 267 equations)

This paper contains 43 sections, 12 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Setting and Notations
  3. Assumptions on the initial wave
  4. Assumptions on the dispersion relation
  5. Asumptions on target rates
  6. Main results
  7. Impulsive control
  8. Linear memory-feedback control
  9. Impulsive control - Proof of Theorem \ref{['Main']}
  10. Preliminary computations
  11. Thermal terms
  12. First thermal term \ref{['thermal1']}
  13. Second and third thermal terms \ref{['thermal2']},\ref{['thermal3']}
  14. Last thermal term \ref{['thermal4']}
  15. Scattering terms
  16. ...and 28 more sections

Key Result

Theorem 2

Consider the wave system governed by dbeta and dalpha along with all the assumptions init0 - W_0, expdecay - positivesigma, monoome - inverome and Passumption - integralF^2bound. For any $\tau>0$ and $O\in L^{1}([0,\tau];\mathscr N)$ we have where

Theorems & Definitions (22)

  • Remark 1
  • Definition 1
  • Theorem 2
  • Remark 3
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Corollary 7
  • Remark 8
  • ...and 12 more