Advances in Computational Mathematics

, Volume 26, Issue 1–3, pp 137–153

Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems

Article

Abstract

We study the numerical approximation of distributed optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Piecewise linear finite elements are used to approximate the control as well as the state. We prove that the L2-error estimates are of order o(h), which is optimal according with the \(C^{0,1}(\overline{\Omega})\) -regularity of the optimal control.

Keywords

optimal control semilinear elliptic equations numerical approximation error estimates 

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References

  1. [1]
    N. Arada, E. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Comput. Optim. Appl. 23 (2002) 201–229. MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    J. Bonnans and H. Zidani, Optimal control problems with partially polyhedric constraints, SIAM J. Control Optim. 37 (1999) 1726–1741. MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    E. Casas, Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints, ESAIM: COCV 8 (2002) 345–374. MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    E. Casas and M. Mateos, Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints, SIAM J. Control Optim. 40 (2002) 1431–1454. MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math. 21 (2002) 67–100. MathSciNetMATHGoogle Scholar
  6. [6]
    E. Casas, M. Mateos and F. Tröltzsch, Error estimates for the numerical approximation of boundary semilinear elliptic control problems, Comput. Optim. Appl. (to appear). Google Scholar
  7. [7]
    E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations (to appear). Google Scholar
  8. [8]
    E. Casas and F. Tröltzsch, Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory, SIAM J. Optim. 13 (2002) 406–431. MATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, Amsterdam, 1978). MATHGoogle Scholar
  10. [10]
    A. Dontchev and W. Hager, The Euler approximation in state constrained optimal control, Math. Comput. 70 (2000) 173–203. MathSciNetGoogle Scholar
  11. [11]
    A. Dontchev and W. Hager, Second-order Runge–Kutta approximations in constrained optimal control, SIAM J. Numer. Anal. 38 (2000) 202–226. MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    R. Falk, Approximation of a class of optimal control problems with order of convergence estimates, J. Math. Anal. Appl. 44 (1973) 28–47. MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Numer. Anal. 13 (1979) 313–328. MATHMathSciNetGoogle Scholar
  14. [14]
    P. Grisvard, Elliptic Problems in Nonsmooth Domains (Pitman, Boston, 1985). MATHGoogle Scholar
  15. [15]
    W. Hager, Multiplier methods for nonlinear optimal control, SIAM J. Numer. Anal. 27 (1990) 1061–1080. MATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    W. Hager, Numerical analysis in optimal control, in: Optimal Control of Complex Structures, International Series of Numerical Mathematics, Vol. 139 (Birkhäuser, Basel, 2001) pp. 83–93. Google Scholar
  17. [17]
    G. Knowles, Finite element approximation of parabolic time optimal control problems, SIAM J. Control Optim. 20 (1982) 414–427. MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    I. Lasiecka, Boundary control of parabolic systems: finite-element approximations, Appl. Math. Optim. 6 (1980) 287–333. MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    I. Lasiecka, Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, SIAM J. Control Optim. 97 (1984) 477–500. MathSciNetCrossRefGoogle Scholar
  20. [20]
    K. Malanowski, C. Büskens and H. Maurer, Convergence of approximations to nonlinear control problems, in: Mathematical Programming with Data Perturbation, ed. A.V. Fiacco (Marcel Dekker, New York, 1997) pp. 253–284. Google Scholar
  21. [21]
    R. McKnight and W. Bosarge, The Ritz–Galerkin procedure for parabolic control problems, SIAM J. Control Optim. 11 (1973) 510–524. MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    P. Raviart and J. Thomas, Introduction à l'analyse numérique des equations aux dérivées partielles (Masson, Paris, 1983). MATHGoogle Scholar
  23. [23]
    J. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints, Discrete Contin. Dynam. Systems 6 (2000) 431–450. MATHMathSciNetCrossRefGoogle Scholar
  24. [24]
    D. Tiba and F. Tröltzsch, Error estimates for the discretization of state constrained convex control problems, Numer. Funct. Anal. Optim. 17 (1996) 1005–1028. MATHMathSciNetGoogle Scholar
  25. [25]
    F. Tröltzsch, Semidiscrete finite element approximation of parabolic boundary control problems-convergence of switching points, in: Optimal Control of Partial Differential Equations II, International Series of Numerical Mathematics, Vol. 78 (Birkhäuser, Basel, 1987) pp. 219–232. Google Scholar
  26. [26]
    F. Tröltzsch, Approximation of nonlinear parabolic boundary problems by the Fourier method-convergence of optimal controls, Optimization 2 (1991) 83–98. Google Scholar
  27. [27]
    F. Tröltzsch, On a convergence of semidiscrete Ritz–Galerkin schemes applied to the boundary control of parabolic equations with non-linear boundary condition, Z. Angew. Math. Mech. 72 (1992) 291–301. MATHMathSciNetGoogle Scholar
  28. [28]
    F. Tröltzsch, Semidiscrete Ritz–Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls, Appl. Math. Optim. 29 (1994) 309–329. MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Dpto. de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de TelecomunicaciónUniversidad de CantabriaSantanderSpain

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