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Optimal Control in a Bounded Domain for Wave Propagating in the Whole Space: Coupling Through Boundary Integral Equations

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Abstract

This paper is concerned with an optimal control problem in a bounded-domain \(\varOmega _0\) under the constraint of a wave equation in the whole space. The problem is regularized and then reformulated as an initial-boundary value problem of the wave equation in a bounded domain \(\varOmega \supset {\overline{\varOmega }}_0\) coupled with a set of boundary integral equations on \(\partial \varOmega \) taking account of wave propagation through the boundary. The well-posedness and stability of the reformulated problem are proved. A fully discrete finite element method is proposed for solving the reformulated problem. In particular, the wave equation in the bounded domain is discretized by an averaged central difference method in time, and the boundary integral equations are discretized in time by using the convolution quadrature generated by the second-order backward difference formula. The finite and boundary element methods are used for spatial discretization of the wave equation and the boundary integral equations, respectively. The stability and convergence of the numerical method are also proved. Finally, the spatial and temporal convergence rates are validated numerically in 2D.

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Funding

The research of W. Gong was supported in part by the Key Research Program of the Chinese Academy of Sciences under Grant XDPB11, the National Key Basic Research Program (No. 2018YFB0704304) and the National Natural Science Foundation of China under Grant 11671391. The work of Buyang Li was partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (GRF Project No. PolyU15300817), and an internal grant at The Hong Kong Polytechnic University (PolyU Project ID: P0031035, Work Programme: ZZKQ). The research of H. Yang was supported in part by the key research projects of general universities in Guangdong Province (Grant No. 2019KZDXM034), and the basic research and applied basic research projects in Guangdong Province (Projects of Guangdong, Hong Kong and Macao Center for Applied Mathematics, Grant No. 2020B1515310018).

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Authors and Affiliations

  1. The State Key Laboratory of Scientific and Engineering Computing, Institute of Computational Mathematics & National Center for Mathematics and Interdisciplinary Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 100190, Beijing, China

    Wei Gong

  2. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Buyang Li

  3. Department of Mathematics, Shantou University, Shantou, 515063, China

    Huanhuan Yang

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  1. Wei Gong
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Gong, W., Li, B. & Yang, H. Optimal Control in a Bounded Domain for Wave Propagating in the Whole Space: Coupling Through Boundary Integral Equations. J Sci Comput 92, 91 (2022). https://doi.org/10.1007/s10915-022-01953-1

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