Finite Spectral-Band Optimal Control of Acoustic Waves via Subwavelength Point-Like Resonant Actuators

Abstract

We study finite-band optimal control of acoustic waves actuated by local clusters of subwavelength resonators. The acoustic problem reduces to a time-domain Foldy-Lax approximation capturing wave-structure interaction. Spectral analysis of the delayed transfer matrix isolates collective scattering resonances corresponding to weakly damped poles $s_{\alpha }^{\epsilon }=-{\gamma }_{\alpha }^{\epsilon }+i{\omega }_{\alpha }^{\epsilon }$ with radiation damping ${\gamma }_{\alpha }^{\epsilon }>0$. Projecting onto a finite band yields the coupled system $\ddot{a}+\Lambda a=C_{\epsilon }\eta$, $\ddot{\eta }+2{\Gamma }_{\epsilon }\dot{\eta }+K_{\epsilon }\eta =u$, where $a$, $\eta$, and $u$ are modal coefficients, microstructural states, and control. For a tracking functional ${\mathcal{J}}_{\mu }$ with regularization $\mu >0$, we prove existence and uniqueness of the optimal control and derive the adjoint system. Our main quantitative result is a resonant source-lifting estimate: if a source profile ${\eta }_r$ is spectrally concentrated in bands $I_{\alpha }$, the input $u_r=({\partial }_t^2+2{\Gamma }_{\epsilon }{\partial }_t+K_{\epsilon }){\eta }_r$ satisfies $\Vert u_r{\Vert }_{L^2(0,T)}^2\le {\sum }_{\alpha }{\left({\sup }_{\nu \in I_{\alpha }}|({\omega }_{\alpha }^{\epsilon }{)}^2+({\gamma }_{\alpha }^{\epsilon }{)}^2-{\nu }^2+2i{\gamma }_{\alpha }^{\epsilon }\nu |\right)}^2\Vert ({\eta }_r{)}_{\alpha }{\Vert }_{{\mathcal{B}}_T(I_{\alpha })}^2$. This provides an upper bound for the optimal value function. At exact matching $\nu ={\omega }_{\alpha }^{\epsilon }$, the multiplier equals $2{\gamma }_{\alpha }^{\epsilon }{\omega }_{\alpha }^{\epsilon }+O(({\gamma }_{\alpha }^{\epsilon }{)}^3)$, showing clustering yields a finite resonant gain governed by the pole's real part. Finally, this attenuation enables finite-band stabilization under an explicit modal coupling condition, with a decay rate proportional to the cluster damping scale.

Keywords. Wave equation; Scattering resonances; Subwavelength resonators; Foldy-Lax hyperbolic system; Spectral projection; Linear-quadratic optimal control; LaSalle/Huang--Prüss principle; Asymptotic stabilization