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| Format: | Preprint |
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2026
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| Online-Zugang: | https://arxiv.org/abs/2605.25978 |
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| _version_ | 1866914600073560064 |
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| author | Mukherjee, Arpan Sini, Mourad |
| author_facet | Mukherjee, Arpan Sini, Mourad |
| contents | 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 $\mathcal{O}(\varepsilon^γ)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_25978 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Resonant Microstructures as Dirac-type Actuators for Acoustic Wave Control Mukherjee, Arpan Sini, Mourad Analysis of PDEs Optimization and Control 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 $\mathcal{O}(\varepsilon^γ)$ 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. |
| title | Resonant Microstructures as Dirac-type Actuators for Acoustic Wave Control |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2605.25978 |