Computational Optimization and Applications

, Volume 63, Issue 3, pp 793–824 | Cite as

Finite element error estimates for an optimal control problem governed by the Burgers equation

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Abstract

We derive a-priori error estimates for the finite-element approximation of a distributed optimal control problem governed by the steady one-dimensional Burgers equation with pointwise box constraints on the control. Here the approximation of the state and the control is done by using piecewise linear functions. With this choice, a superlinear order of convergence for the control is obtained in the \(L^2\)-norm; moreover, under a further assumption on the regularity structure of the optimal control this error estimate can be improved to \(h^{3/2}\), extending the results in Rösch (Optim. Methods Softw. 21(1): 121–134, 2006). The theoretical findings are tested experimentally by means of numerical examples.

Keywords

Optimal control Burgers equation Finite element approximation Piecewise linear Error estimates 

Mathematics Subject Classification

35Q53 49K20 49J20 80M10 49N05 65N12 41A25 

References

  1. 1.
    Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23, 201–229 (2002)CrossRefMathSciNetMATHGoogle Scholar
  2. 2.
    Burns, J.A., Kang, S.: A control problem for Burgers’ equation with bounded input/output. Nonlinear Dyn. 2(4), 235–262 (1991)CrossRefGoogle Scholar
  3. 3.
    Burns, J.A., Kang, S.: A stabilization problem for burgers equation with unbounded control and observation. Estimation and Control of Distributed Parameter Systems, pp. 51–72. Springer, Berlin (1991)CrossRefGoogle Scholar
  4. 4.
    Casas, E.: Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems. Adv. Comput. Math. 26, 137–156 (2007)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. J. Comput. Appl. Math. 21, 67–100 (2002)MathSciNetMATHGoogle Scholar
  6. 6.
    Casas, E., Mateos, M., Raymond, J.-P.: Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier-Stokes equations. SIAM J. Control Optim. 46(3), 952–982 (2007). (electronic)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Ciarlet, P.G., Lions, L.L.: Handbook of Numerical Analysis, Part I. Finite Element Methods, vol. II. North-Holland, Amsterdam (1991)Google Scholar
  8. 8.
    de los Reyes, J.C., Kunisch, K.: A comparison of algorithms for control constrained optimal control of the burgers equation. Calcolo 41(4), 203–225 (2004)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    de los Reyes, J.C., Meyer, C., Vexler, B.: Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybern. 37(2), 251–284 (2008)MATHGoogle Scholar
  10. 10.
    Casas, E., de los Reyes, J.C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19(2), 616–643 (2008)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)Google Scholar
  12. 12.
    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, vol. 5. Springer, Berlin (2012)Google Scholar
  13. 13.
    Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Rösch, A.: Error estimates for linear-quadratic control problems with control constraints. Optim. Methods Softw. 21(1), 121–134 (2006)CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Temam, R.: Navier–Stokes Equations and Nonlinear Functional Analysis, vol. 66. SIAM, Philadelphia (1995)CrossRefMATHGoogle Scholar
  16. 16.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations. Graduate Studies in Mathematics. American Mathematical Society, Providence (2010)CrossRefMATHGoogle Scholar
  17. 17.
    Volkwein, S.: Mesh independence of an augmented lagrangean-SQP method in Hilbert spaces and control problems for the Burgers equation. Ph.D. Thesis, Technische Universität Berlin (1997)Google Scholar
  18. 18.
    Zowe, J., Kurcyusz, S.: Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62 (1979)CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.ModeMat: Research Center on Mathematical Modeling, Departamento de MatemáticaEscuela Politécnica NacionalQuitoEcuador

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