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Spectral Method Approximation of Flow Optimal Control Problems with H1-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)
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Abstract

In this paper, we consider an optimal control problem governed by Stokes equations with H1-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
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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References

[1] Benedix, O. and Vexler, B., A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Comput. Optim. Appl., 44(1) (2009), pp. 3–25.CrossRefGoogle Scholar
[2] Bergounioux, M. and Kunisch, K., On the structure of Lagrange multipliers for state-constrained optimal control problems, Syst. Control Lett., 48 (2003), pp. 169–176.CrossRefGoogle Scholar
[3] Bergounioux, M. and Kunisch, K., Augmented Lagrangian techniques for elliptic state constrained optimal control problems, SIAM J. Control Optim., 35 (1997), pp. 1524–1543.CrossRefGoogle Scholar
[4] Bergounioux, M. and Kunisch, K., Primal-dual strategy for state-constrained optimal control problems, Comput. Optim. Appl., 22 (2002), pp. 193–224.CrossRefGoogle Scholar
[5] Barbu, V. and Precupanu, T., Convexity and Optimization in Banach Spaces, Springer, 2012.CrossRefGoogle Scholar
[6] Casas, E., Control of an elliptic problem with pointwise state constraints, SIAM J. Control Optim., 24(6) (1986), pp. 1309–1318.CrossRefGoogle Scholar
[7] Casas, E., Boundary control of semilinear elliptic equations with pointwise state constraints, SIAM J. Control Optim., 31(4) (1993), pp. 993–1006.CrossRefGoogle Scholar
[8] Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988.CrossRefGoogle Scholar
[9] Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, Berlin, 2006.CrossRefGoogle Scholar
[10] Canuto, C., Hussaini, M.Y., Quarteroni, A., and Zang, T.A., Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Springer-Verlag, Berlin, 2007.CrossRefGoogle Scholar
[11] Clarke, F.H., Optimization and Nonsmooth Analysis, John Wiley Sons, Amsterdam, 1983.Google Scholar
[12] Cherednichenko, S. and Rösch, A., Error estimates for the discretization of elliptic control problems with pointwise control and state constraints, Comput. Optim. Appl., 44(1) (2009), pp. 27–55.CrossRefGoogle Scholar
[13] Chen, Y., Yi, N., and Liu, W.B., A Legendre Galerkin spectral method for optimal control problems governed by elliptic equations, SIAM J. Numer. Anal., 46(5) (2008), pp. 2254–2275.CrossRefGoogle Scholar
[14] De Los Reyes, J.C. and Griesse, R., State-constrained optimal control of the three-dimensionl stationary Navier-Stokes equations, J. Math. Anal. Appl., 343 (2008), pp. 257–272.CrossRefGoogle Scholar
[15] De Los Reyes, J.C. and Kunisch, K., A semi-smooth Newton method for regularized state-constrained optimal control of the Navier-Stokes equations, Comput., 78 (2006), pp. 287–309.CrossRefGoogle Scholar
[16] Gunzburger, M.D. and Hou, L.S., Finite-dimensional approximation of a class of constrained nonlinear optimal control problems, SIAM J. Control Optim., 34(3) (1996), pp. 1001–1043.CrossRefGoogle Scholar
[17] Gong, W. and Yan, N.N., A mixed finite element scheme for optimal control problems with pointwise state constraints, J. Sci. Comput., 46 (2011), pp. 182–203.CrossRefGoogle Scholar
[18] Glowinski, R., Lions, J.L., and Trémolières, R., Numerical analysis of variational inequalities, Amsterdam, North-Holland, 1981.Google Scholar
[19] Hoppe, R.H.W. and Kieweg, M., A posteriori error estimation of finite element approximations of pointwise state constrained distributed control problems, J. Numer. Math., 17(3) (2009), pp. 219–244.CrossRefGoogle Scholar
[20] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[21] Liu, W. B., Gong, W. and Yan, N., A new finite element approximation of a state-constrained optimal control problem, J. Comp. Math., 27(1) (2009), pp. 97–114.Google Scholar
[22] Liu, W. B., Yang, D. P., Yuan, L., and MA, C. Q., Finite element approximations of an optimal control problem with integral state constraint, SIAM J. Numer. Anal., 48(3) (2010), pp. 1163–1185.CrossRefGoogle Scholar
[23] Liu, W. B. and Yan, N., A posteriori error estimates for control problems governed by Stokes equations, SIAM J. Numer. Anal., 40(5) (2002), pp. 1850–1869.CrossRefGoogle Scholar
[24] Liu, W.B. and Yan, N., Adaptive finite element methods for optimal control governed by PDEs, Science Press, Beijing, 2008.Google Scholar
[25] Meidner, D., Rannacher, R., and Vexler, B., A priori error estimates for finite element discretizations of parabolic optimization problems with pointwise state constraints in time, SIAM J. Control Optim., 49(5) (2011), pp. 1961–1997.CrossRefGoogle Scholar
[26] Niu, H. F. and Yang, D. P., Finite element analysis of optimal control problem governed by Stokes equations with L2-norm state-constriants, J. Comput. Math., 29(5) (2011), pp. 589–604.Google Scholar
[27] Neittaanmaki, P. and Tiba, D., Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, Marcel Dekker, New York, 1994 Google Scholar
[28] Pironneau, O., Optimal Shape Design for Elliptic Systems, Springer-Verlag, Berlin, 1984.CrossRefGoogle Scholar
[29] Shen, J., On fast direct poisson solver, inf-sup constant and iterative Stokes solver by Legendre-Galerkin method, J. Comput. Phys., 116(1) (1995), pp. 184–188.CrossRefGoogle Scholar
[30] Shen, J. and Tang, T., Spectral and High-Order Methods with Applications, Science Press, Beijing, 2006.Google Scholar
[31] Yuan, L. and Yang, D. P., A posteriori error estimate of optimal control problem of PDE with integral constraint for state, J. Comput. Math., 27(4) (2009), pp. 525–542.Google Scholar
[32] Zhou, J. W. and Yang, D. P., Spectral mixed Galerkin method for state constrained optimal control problem governed by the first bi-harmonic equation, Int. J. Comput. Math., 88(14) (2011), pp. 2988–3011.CrossRefGoogle Scholar