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Legendre–Galerkin spectral methods for optimal control problems with integral constraint for state in one dimension

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Abstract

In this paper, we investigate the optimal control problems governed by elliptic equations with integral constraint for state variable in one dimension by Legendre–Galerkin spectral methods. We deduce optimal conditions of the optimal control problems. Meanwhile, we obtain an a priori error estimate and a posteriori error estimator. Furthermore, we obtain an explicit formula of the a posteriori error estimator by orthogonal properties of Legendre polynomials. Finally, we present numerical examples to confirm our analytical results.

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Acknowledgments

This work is partly supported by NSFC (Nos. 11201212, 11301252), CSC (No. 20140837045), Promotive Research Fund for Excellent Young and Middle-aged Scientists of Shandong Province (No. BS2012DX004), AMEP, LYDX2013BS054 and Special Funds for Doctoral Authorities of Linyi University.

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Correspondence to Jianwei Zhou.

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Zhou, J., Yang, D. Legendre–Galerkin spectral methods for optimal control problems with integral constraint for state in one dimension. Comput Optim Appl 61, 135–158 (2015). https://doi.org/10.1007/s10589-014-9700-x

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  • DOI: https://doi.org/10.1007/s10589-014-9700-x

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