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A priori error analysis of mixed methods for nonlinear quadratic optimal control problems

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Abstract

In this paper, we study an a priori error analysis for the quadratic optimal control problems governed by nonlinear elliptic partial differential equations using mixed finite element methods. The state and the costate are approximated by the lowest order Raviart-Thomas mixed finite element spaces, and the control is approximated by piecewise constant functions. A priori error estimates for the mixed finite element approximation of nonlinear optimal control problems are obtained. Some numerical examples are presented to confirm our theoretical results.

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References

  1. N. Arada, E. Casas, and F. Tröltzsch, Error Estimates for the Numerical Approximation of a Semilinear Elliptic Control Problem, Comp. Optim. Appl. 23, 201 (2002).

    Article  MATH  Google Scholar 

  2. P. Clement, Approximation by Finite Element Funtions Using Local Regularization, RAIRO Ser. Rougr Anal. Numer. 2, 77 (1995).

    Google Scholar 

  3. Y. Kwon and F. A. Milner, L ∞-error Estimates for Mixed Methods for Semilinear Second-order Elliptic Equations, SIAMJ. Numer. Anal. 46, 25 (1988).

    MathSciNet  Google Scholar 

  4. M. A. Ignatieva and A. V. Lapin, Mixed Hybrid Finite Element Scheme for Stefan Problem with Prescribed Convection, Lobachevskii J. Math. 13, 15 (2003).

    MATH  MathSciNet  Google Scholar 

  5. Z. L. Lu and H. W. Zhang, V Cycle Multi-grid-cycle Method of Viscoelastic Fluid Flow Obeying an Oldroyd B Type Constitutive Law, Numer. Anal. Appl. 11, 83 (2008).

    Google Scholar 

  6. R. F. Kadyrov, E. Laitinen, and A. V. Lapin, Using Explicit Schemes for Control Problems in Continuous Casting Process, Lobachevskii J. Math. 13, 25 (2005).

    MathSciNet  Google Scholar 

  7. W. B. Liu and J. W. Barrett, Quasi-norm Error Bounds for the Finite Element Approximation Some Degenerate Quasilinear Elliptic Equations and Variational Inequalities, Math. Modelling Numer. Anal. 28, 725 (1994).

    MATH  MathSciNet  Google Scholar 

  8. Y. P. Chen and W. B. Liu, Error Estimates and Superconvergence of Mixed Finite Elements for Quadratic Optimal Control, Internat. J. Numer. Anal. Modeling 3, 311 (2006).

    MATH  Google Scholar 

  9. F. S. Falk, Approximation of a Class of Optimal Control Problems with Order of Convergence Estimates, J. Math. Anal. Appl. 44, 28 (1973).

    Article  MATH  MathSciNet  Google Scholar 

  10. M. D. Gunzburger and S. L. Hou, Finite Dimensional Approximation of a Class of Constrained Nonlinear Control Problems, SIAM J. Control Optim. 34, 1001 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  11. Y. P. Chen, N. Y. Yi, and W. B. Liu, A Legendre Galerkin Spectral Method for Optimal Control Problems Governed by Elliptic Equations, SIAM J. Numer. Anal. (in press).

  12. F. A. Miliner, Mixed Finite Element Methods for Quasilinear Second-order Elliptic Problems, Math. Comp. 44, 303 (1985).

    Article  MathSciNet  Google Scholar 

  13. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equtions (Springer, Berlin, 1971), p. 149.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, **angtan University, **angtan, 411105, China

    Z. L. Lu

  2. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

    Y. P. Chen

  3. College of Math and Computing Science, Changsha University of Science and Technology, Changsha, 410076, China

    H. W. Zhang

Authors
  1. Z. L. Lu
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  2. Y. P. Chen
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  3. H. W. Zhang
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Correspondence to Z. L. Lu.

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Submitted by A.V. Lapin

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Lu, Z.L., Chen, Y.P. & Zhang, H.W. A priori error analysis of mixed methods for nonlinear quadratic optimal control problems. Lobachevskii J Math 29, 164–174 (2008). https://doi.org/10.1134/S1995080208030074

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  • DOI: https://doi.org/10.1134/S1995080208030074

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