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Spectral Method Approximation of Flow Optimal Control Problems with H 1-Norm State Constraint

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Existence theories

Published online by Cambridge University Press:  20 June 2017

Yanping Chen  and
Fenglin Huang
Yanping Chen*
Affiliation:
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
Fenglin Huang*
Affiliation:
School of Mathematics and Statistics, Xinyang Normal University, No.237 Nanhu Road, Shihe District, Xinyang 464000, China
*
*Corresponding author. Email addresses:yanpingchen@scnu.edu.cn (Y. P. Chen), hfl_937@sina.com (F. L. Huang)
*Corresponding author. Email addresses:yanpingchen@scnu.edu.cn (Y. P. Chen), hfl_937@sina.com (F. L. Huang)

Abstract

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In this paper, we consider an optimal control problem governed by Stokes equations with H 1-norm state constraint. The control problem is approximated by spectral method, which provides very accurate approximation with a relatively small number of unknowns. Choosing appropriate basis functions leads to discrete system with sparse matrices. We first present the optimality conditions of the exact and the discrete optimal control systems, then derive both a priori and a posteriori error estimates. Finally, an illustrative numerical experiment indicates that the proposed method is competitive, and the estimator can indicate the errors very well.

Keywords

Optimal controlstate constraintstokes equationslegendre polynomialsspectral method

MSC classification

Secondary: 49J20: Optimal control problems involving partial differential equations 65K15: Numerical methods for variational inequalities and related problems 65M12: Stability and convergence of numerical methods 65M70: Spectral, collocation and related methods

Information

Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 614 - 638
Copyright
Copyright © Global-Science Press 2017 

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