Table of Contents
ISRN Applied Mathematics
Volume 2014, Article ID 579047, 12 pages
http://dx.doi.org/10.1155/2014/579047
Research Article

Superconvergence for General Convex Optimal Control Problems Governed by Semilinear Parabolic Equations

1Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, China
2School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631, China

Received 16 September 2013; Accepted 12 December 2013; Published 10 February 2014

Academic Editors: M. Braack, Y. M. Cheng, and W. Yeih

Copyright © 2014 Yongquan Dai and Yanping Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. W. Alt, “On the approximation of infinite optimization problems with an application to optimal control problems,” Applied Mathematics and Optimization, vol. 12, no. 1, pp. 15–27, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  2. W. Alt and U. Mackenroth, “Convergence of finite element approximations to state constrained convex parabolic boundary control problems,” SIAM Journal on Control and Optimization, vol. 27, no. 4, pp. 718–736, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Y. Chen, “Superconvergence of quadratic optimal control problems by triangular mixed finite element methods,” International Journal for Numerical Methods in Engineering, vol. 75, no. 8, pp. 881–898, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Y. Chen, “Superconvergence of mixed finite element methods for optimal control problems,” Mathematics of Computation, vol. 77, no. 263, pp. 1269–1291, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Y. Chen and Y. Dai, “Superconvergence for optimal control problems governed by semi-linear elliptic equations,” Journal of Scientific Computing, vol. 39, no. 2, pp. 206–221, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  6. Y. Chen and W. Liu, “Error estimates and superconvergence of mixed finite element for quadratic optimal control,” International Journal of Numerical Analysis and Modeling, vol. 3, no. 3, pp. 311–321, 2006. View at Google Scholar · View at MathSciNet
  7. Y. Chen and W. Liu, “A posteriori error estimates for mixed finite element solutions of convex optimal control problems,” Journal of Computational and Applied Mathematics, vol. 211, no. 1, pp. 76–89, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  8. D. A. French and J. T. King, “Approximation of an elliptic control problem by the finite element method,” Numerical Functional Analysis and Optimization, vol. 12, no. 3-4, pp. 299–314, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  9. L. S. Hou and J. C. Turner, “Analysis and finite element approximation of an optimal control problem in electrochemistry with current density controls,” Numerische Mathematik, vol. 71, no. 3, pp. 289–315, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  10. G. Knowles, “Finite element approximation of parabolic time optimal control problems,” Society for Industrial and Applied Mathematics, vol. 20, no. 3, pp. 414–427, 1982. View at Publisher · View at Google Scholar · View at MathSciNet
  11. I. Lasiecka, “Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions,” SIAM Journal on Control and Optimization, vol. 22, no. 3, pp. 477–500, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  12. R. S. McKnight and W. E. Bosarge, Jr., “The Ritz-Galerkin procedure for parabolic control problems,” SIAM Journal on Control and Optimization, vol. 11, pp. 510–524, 1973. View at Google Scholar · View at MathSciNet
  13. D. Meidner and B. Vexler, “A priori error estimates for space-time finite element discretization of parabolic optimal control problems. I. Problems without control constraints,” SIAM Journal on Control and Optimization, vol. 47, no. 3, pp. 1150–1177, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  14. D. Meidner and B. Vexler, “A priori error estimates for space-time finite element discretization of parabolic optimal control problems. II. Problems with control constraints,” SIAM Journal on Control and Optimization, vol. 47, no. 3, pp. 1301–1329, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  15. P. G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publishing, Amsterdam, The Netherlands, 1978. View at MathSciNet
  16. J. Haslinger and P. Neittaanmäki, Finite Element Approximation for Optimal Shape Design, John Wiley & Sons, Chichester, UK, 1988. View at MathSciNet
  17. P. Neittaanmäki and D. Tiba, Optimal control of Nonlinear Parabolic Systems, vol. 179, Marcel Dekker, New York, NY, USA, 1994. View at MathSciNet
  18. D. Tiba, Lectures on the Optimal Control of Elliptic Equations, University of Jyvaskyla Press, Jyvaskyla, Finland, 1995.
  19. Q. Li and H. Du, “Lp error estimates and superconvergence for finite element approximation for nonlinear parabolic problems,” Journal of the Korean Society for Industrial and Applied Mathematics, vol. 4, pp. 67–77, 2000. View at Google Scholar
  20. C. Hou, Y. Chen, and Z. Lu, “Superconvergence property of finite element methods for parabolic optimal control problems,” Journal of Industrial and Management Optimization, vol. 7, no. 4, pp. 927–945, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  21. K. Chrysafinos, M. D. Gunzburger, and L. S. Hou, “Semidiscrete approximations of optimal Robin boundary control problems constrained by semilinear parabolic PDE,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 891–912, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. F. Tröltzsch, “Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems—strong convergence of optimal controls,” Applied Mathematics and Optimization, vol. 29, no. 3, pp. 309–329, 1994. View at Publisher · View at Google Scholar · View at MathSciNet
  23. J. L. Lions and E. Magenes, Non Homogeneous Boundary Value Problems and Applications, vol. 181, Edited by B. Grandlehre, Springer, 1972.
  24. J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, New York, NY, USA, 1971. View at MathSciNet
  25. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations, vol. 5, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  26. R. Li, W. Liu, and N. Yan, “A posteriori error estimates of recovery type for distributed convex optimal control problems,” Journal of Scientific Computing, vol. 33, no. 2, pp. 155–182, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  27. W. Liu and D. Tiba, “Error estimates in the approximation of optimization problems governed by nonlinear operators,” Numerical Functional Analysis and Optimization, vol. 22, no. 7-8, pp. 953–972, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  28. Y. Huang, R. Li, W. Liu, and N. Yan, “Adaptive multi-mesh finite element approximation for constrained optimal control problems,” SIAM Journal on Control and Optimization. In press.
  29. R. Li and W. B. Liu, http://dsec.pku.edu.cn/~rli/software.php#AFEPack.