Resonant Microstructures as Dirac-type Actuators for Acoustic Wave Control

Abstract

We study interior control of the acoustic wave equation via effective point sources generated by a finite cluster of resonant perturbations (modeling acoustic subwavelength bubbles). At the abstract level, after localizing the whole-space dynamics to a large auxiliary observation domain, we consider a Dirichlet spectral formulation of the wave equation with finitely many point actuators located at prescribed interior points. Restricting to a finite spectral band of Dirichlet eigenfrequencies, we prove that, under a natural full-rank condition on the associated coupling matrix, arbitrary trajectories on the corresponding spectral subspace can be realized, with quantitative bounds on the control cost in terms of spectral-band geometry and actuator placement. We then show that these ideal actuators can be realized by clusters of small, high-contrast bubbles. Using a time-domain asymptotic expansion, the scattered field is represented as a superposition of retarded monopoles whose amplitudes satisfy a finite-dimensional delayed hyperbolic system. In the Laplace domain, this induces a transfer operator whose pole structure encodes the Minnaert resonance with a collective attenuation. We prove that the associated actuator map is ill-conditioned away from resonance, whereas, under a cluster-level transducer accessibility condition linking the incident fields to the dominant cluster channels, it admits a bounded right inverse on suitable Minnaert bands. Consequently, one obtains spectral tracking of the wave field with error $\mathcal{O}(\varepsilon^\gamma)$ as the bubble size $\varepsilon \to 0$. Keywords. Wave equation, Dirac actuators, Trajectory tracking control, Resonant perturbations, Kato's analytic perturbation, Perron-Frobenius spectrum, Minnaert resonances, Actuation map, Toeplitz matrix.