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

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 $V^{\mathrm{\prime }}$ 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.