Abstract
We consider initial boundary-value problem for acoustic equation in the time space cylinder Ω × (0; 2T) with unknown variable speed of sound, zero initial data, and mixed boundary conditions. We assume that (Neumann) controls are located at some part Σ Ω [0; T]; Σ ⊂ 𝜕Ω of the lateral surface of the cylinder Ω × (0; T). The domain of observation is Σ × [0; 2T]; and the pressure on another part (𝜕ΩnΣ) × [0; 2T]) is assumed to be zero for any control. We prove the approximate boundary controllability for functions from the subspace V ⊂ H1(Ω) whose traces have vanished on Σ provided that the observation time is 2T more than two acoustic radii of the domain Ω. We give an explicit procedure for solving Boundary Control Problem (BCP) for smooth harmonic functions from V (i.e., we are looking for a boundary control f which generates a wave uf such that uf (., T) approximates any prescribed harmonic function from V ). Moreover, using the Friedrichs–Poincaré inequality, we obtain a conditional estimate for this BCP. Note that, for solving BCP for these harmonic functions, we do not need the knowledge of the speed of sound.
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Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 15, No. 2, pp. 251–263, January–March, 2018.
This work was supported by the Volkswagen Foundation project “Modeling, Analysis and Approximation Theory toward Applications in Tomography and Inverse Problem”. The authors would like also to thank M. Belishev, V. Derkach, and T. Fastovska for useful discussions and valuable remarks.
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Pestov, L., Strelnikov, D. Approximate controllability of the wave equation with mixed boundary conditions. J Math Sci 239, 75–85 (2019). https://doi.org/10.1007/s10958-019-04289-8
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DOI: https://doi.org/10.1007/s10958-019-04289-8