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 $s_α^ε=-γ_α^ε+iω_α^ε$ with radiation damping $γ_α^ε>0$. Projecting onto a finite band yields the coupled system $\ddot{a}+Λa=C_εη$, $\ddotη+2Γ_ε\dotη+K_εη=u$, where $a$, $η$, and $u$ are modal coefficients, microstructural states, and control. For a tracking functional $\mathcal{J}_μ$ with regularization $μ>0$, 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 $η_r$ is spectrally concentrated in bands $I_α$, the input $u_r=(\partial_t^2+2Γ_ε\partial_t+K_ε)η_r$ satisfies $\|u_r\|_{L^2(0,T)}^2 \le \sum_α\left(\sup_{ν\in I_α} |(ω_α^ε)^2+(γ_α^ε)^2-ν^2 +2iγ_α^εν|\right)^2 \|(η_r)_α\|_{\mathcal{B}_T(I_α)}^2$. This provides an upper bound for the optimal value function. At exact matching $ν=ω_α^ε$, the multiplier equals $2γ_α^εω_α^ε+O((γ_α^ε)^3)$, 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.