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Feedback Stabilization and Tracking for Heat Equations Using Thermo-Plasmonic Nanoparticles as Actuators

Arpan Mukherjee, Sérgio S. Rodrigues, Mourad Sini

Abstract

We propose a feedback strategy to track prescribed heat profiles using plasmonic nanoparticles as actuators. Starting from a thermo--plasmonic Maxwell--heat model, we use a time-domain discrete effective description in which the generated heat is approximated by a superposition of heat kernels centered at particle locations with amplitudes governed by a coupled Volterra system. We recast this dynamics as a heat equation on a bounded domain with finitely many point actuators and design a tracking feedback based on pointwise evaluations of $\mathcal A^{-1}y$, where $\mathcal A=I-A_0$ and $A_0$ is the Neumann diffusion operator. Working in the natural $V'$ setting with $V=D(\mathcal A)$, we prove exponential stabilization of the tracking error via distribution-actuator theory. For non-equilibrium reference profiles, we add a constant feedforward term and a low-mode fixed-point pre-compensation on $X_N$, ensuring exact steady matching on $X_N$ and an explicit bound on the residual tail mismatch.

Feedback Stabilization and Tracking for Heat Equations Using Thermo-Plasmonic Nanoparticles as Actuators

Abstract

We propose a feedback strategy to track prescribed heat profiles using plasmonic nanoparticles as actuators. Starting from a thermo--plasmonic Maxwell--heat model, we use a time-domain discrete effective description in which the generated heat is approximated by a superposition of heat kernels centered at particle locations with amplitudes governed by a coupled Volterra system. We recast this dynamics as a heat equation on a bounded domain with finitely many point actuators and design a tracking feedback based on pointwise evaluations of , where and is the Neumann diffusion operator. Working in the natural setting with , we prove exponential stabilization of the tracking error via distribution-actuator theory. For non-equilibrium reference profiles, we add a constant feedforward term and a low-mode fixed-point pre-compensation on , ensuring exact steady matching on and an explicit bound on the residual tail mismatch.
Paper Structure (45 sections, 30 theorems, 201 equations)

This paper contains 45 sections, 30 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Thermo-plasmonic heating and effective medium theory
  3. Heat-profile tracking and point actuators
  4. Objective and main contributions
  5. Thermo-plasmonic model and discrete effective description
  6. Coupled Maxwell--heat model in $\mathbb{R}^3$
  7. Discrete thermo-plasmonic effective model
  8. Plasmonic resonances and actuation strength
  9. Heat equation with point actuators on a bounded domain
  10. Physical model and abstract reformulation
  11. Dirac actuators and low-regularity well-posedness
  12. Feedback design and stability analysis
  13. Closed-loop well-posedness
  14. Assumptions and feedback law
  15. Dirac-actuator stabilization and tracking in $V'$
  16. ...and 30 more sections

Key Result

Theorem 2.1

Assume the scaling and regularity hypotheses of CaoMukherjeeSini2025, in particular the subwavelength, separation, and high-contrast conditions on $(c_p,\gamma_p,\varepsilon_p)$ and the nanoparticles $D_i = z_i + \delta B_i$. Then there exist functions such that the temperature difference $w=u-u^{(0)}$ satisfies, for $x\in\mathbb{R}^3\setminus D$ and $t\in(0,T)$, where $\alpha_i:=\gamma_{p,i}-\g

Theorems & Definitions (76)

  • Theorem 2.1: Discrete thermo-plasmonic effective model
  • Remark 2.2
  • Lemma 3.1: Restriction from $\mathbb{R}^3$ to a bounded control domain
  • proof : Proof sketch
  • Remark 3.2
  • Proposition 3.3: Boundedness of $B$ into $D(\mathcal{A})'$
  • proof
  • Proposition 3.4: Well-posedness for Dirac-forced heat equation
  • proof
  • Definition 4.1: Observation operator
  • ...and 66 more