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An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem

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Abstract

This paper investigates the solution of a parameter identification problem associated with the two-dimensional heat equation with variable diffusion coefficient. The singularity of the diffusion coefficient results in a nonlinear inverse problem which makes theoretical analysis rather difficult. Using an optimal control method, we formulate the problem as a minimization problem and prove the existence and uniqueness of the solution in weighted Sobolev spaces. The necessary conditions for the existence of the minimizer are also given. The results can be extended to more general parabolic equations with singular coefficients.

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Acknowledgements

We would like to thank the anonymous referees for their valuable comments and helpful suggestions to improve the earlier version of the paper. The first author was supported by the National Natural Science Foundation of China (No. 11061018) and the Long Yuan Young Creative Talents Support Program (No. 252003). The second author was supported by the Strategic Research Grant of the City University of Hong Kong (No. 7002670). The third author was supported by the National Natural Science Foundation of China (No. 11261029), the Young Foundation of Lanzhou Jiaotong University (No. 2011028), and the Joint Funds of NSF of Gansu Province of China (No. 1212RJZA043).

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Authors and Affiliations

  1. Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, People’s Republic of China

    Zui-Cha Deng, Y.-C. Hon & Liu Yang

  2. Department of Mathematics, City University of Hong Kong SAR, Hong Kong, People’s Republic of China

    Y.-C. Hon

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  1. Zui-Cha Deng
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  2. Y.-C. Hon
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  3. Liu Yang
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Correspondence to Liu Yang.

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Deng, ZC., Hon, YC. & Yang, L. An Optimal Control Method for Nonlinear Inverse Diffusion Coefficient Problem. J Optim Theory Appl 160, 890–910 (2014). https://doi.org/10.1007/s10957-013-0302-z

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  • DOI: https://doi.org/10.1007/s10957-013-0302-z

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