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

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 $\|u_r\|_{L^2(0,T)}^2 \le \sum_\alpha \left(\sup_{\nu\in I_\alpha} |(\omega_\alpha^\epsilon)^2+(\gamma_\alpha^\epsilon)^2-\nu^2 +2i\gamma_\alpha^\epsilon\nu|\right)^2 \|(\eta_r)_\alpha\|_{\mathcal{B}_T(I_\alpha)}^2$.

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.