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FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints

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Abstract

This paper studies the convergence and error estimates of approximate solutions to an optimal control problem governed by semilinear elliptic equations with non-convex cost function and non-convex mixed pointwise constraints, and unbounded constraint set. We discretize the optimal control problems by the finite element method in order to obtain a sequence of mathematical programming problems in finite-dimensional spaces. We show that under certain conditions, the optimal solutions of the obtained mathematical programming problems converge to an optimal solution of the original problem. In particular, if the original problem satisfies the so-called no-gap second-order conditions, then some error estimates of approximate solutions are obtained.

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References

  1. Arada, N., Casas, E., Tröltzsch, F.: Error estimate for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23, 201–229 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  3. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2010)

    Google Scholar 

  4. Casas, E., Tröltzsch, F.: Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control Cybern. 31(3), 695–712 (2002)

    MathSciNet  MATH  Google Scholar 

  5. Casas, E., Mateos, M.: Uniform convergence of the FEM. Applications to state constrained control problems. Comput. Appl. Math. 21, 67–100 (2002)

    MathSciNet  MATH  Google Scholar 

  6. Casas, E., Raymond, J.-P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45, 1586–1611 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casas, E., Tröltzsch, F.: First and second-order optimality conditions for a class optimal control problems with quasilinear elliptic equations. SIAM J. Control Optim. 48, 688–718 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Casas, E., Tröltzsch, F.: Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations. ESAIM COCV 17, 771–800 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Casas, E., Tröltzsch, F.: A general theorem on error estimates with application to a quasilinear elliptic optimal control problem. Comput. Optim. Appl. 53, 173–206 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cherednichenko, S., Rösch, A.: Error estimates for the discretization of elliptic control problems with pointwise control and state constraints. Comput. Optim. Appl. 44, 27–77 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978)

    MATH  Google Scholar 

  12. Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hinze, M., Meyer, C.: Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. Comput. Optim. Appl. 46, 487–510 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hoppe, R.H.W., Kieweg, M.: Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems. Comput. Optim. Appl. 46, 511–533 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kien, B.T., Nhu, V.H., Son, N.H.: Second-order optimality conditions for a semilinear elliptic optimal control problem with mixed pointwise constraints. Set-Valued Var. Anal. 25, 177–210 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, Hoboken (1989)

    MATH  Google Scholar 

  17. Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43, 970–985 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meyer, C.: Error estimates for the finite-element approximation of an elliptic control problem with pointwise state and control constraints. Control Cybern. 37, 53–83 (2008)

    MathSciNet  Google Scholar 

  19. Merino, P., Tröltzsch, F., Boris Vexler, B.: Error estimates for finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space. ESAIM M2AN 44, 167–188 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Neitzel, I., Pfefferer, J., Rösch, A.: Finite element discretization of state-constraint elliptic optimal control problem with semilinear state equation. SIAM J. Control Optim. 53, 874–904 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nhu, V.H., Tuan, N.Q., Giang, N.B., Huong, N.T.T.: Continuity regularity of optimal control solutions to distributed and boundary semilinear elliptic optimal control problems with mixed pointwise control-state constraints. J. Math. Anal. Appl. 512, 126139 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Rösch, A., Steinig, S.: A priori error estimates for a state-constrained elliptic optimal control problem. ESAIM M2AN 46, 1107–1120 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rösch, A., Wachsmuth, D.: Semi-smooth Newton method for an optimal control problem with control and mixed control-state constraints. Optim. Methods Softw. 26, 169–186 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Rösch, A., Tröltzsch, F.: On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46, 1098–1115 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Temam, R.: Navier–Stokes Equations. North Holland Publishing Company, Amsterdam (1979)

    MATH  Google Scholar 

  26. Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory Method and Applications. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

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Acknowledgements

This research was funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.01\(-\)2019.308. A part of this paper was completed at Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors would like to thank VIASM for their financial support and hospitality. The authors would like to thank the anonymous referees for their useful suggestions and comments which improved the manuscript greatly.

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Correspondence to Bui Trong Kien.

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Kien, B.T., Rösch, A., Son, N.H. et al. FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints. J Optim Theory Appl 197, 130–173 (2023). https://doi.org/10.1007/s10957-023-02187-3

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  • DOI: https://doi.org/10.1007/s10957-023-02187-3

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