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FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints

Author

Listed:
  • Bui Trong Kien

    (Vietnam Academy of Science and Technology)

  • Arnd Rösch

    (University of Duisbug-Essen)

  • Nguyen Hai Son

    (Hanoi University of Science and Technology)

  • Nguyen Van Tuyen

    (Hanoi Pedagogical University 2)

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.

Suggested Citation

  • Bui Trong Kien & Arnd Rösch & Nguyen Hai Son & Nguyen Van Tuyen, 2023. "FEM for Semilinear Elliptic Optimal Control with Nonlinear and Mixed Constraints," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 130-173, April.
    • Handle: RePEc:spr:joptap:v:197:y:2023:i:1:d:10.1007_s10957-023-02187-3
    DOI: 10.1007/s10957-023-02187-3
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    References listed on IDEAS

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    1. M. Hinze & C. Meyer, 2010. "Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems," Computational Optimization and Applications, Springer, vol. 46(3), pages 487-510, July.
    2. R. Hoppe & M. Kieweg, 2010. "Adaptive finite element methods for mixed control-state constrained optimal control problems for elliptic boundary value problems," Computational Optimization and Applications, Springer, vol. 46(3), pages 511-533, July.
    3. Eduardo Casas & Fredi Tröltzsch, 2012. "A general theorem on error estimates with application to a quasilinear elliptic optimal control problem," Computational Optimization and Applications, Springer, vol. 53(1), pages 173-206, September.
    4. S. Cherednichenko & A. Rösch, 2009. "Error estimates for the discretization of elliptic control problems with pointwise control and state constraints," Computational Optimization and Applications, Springer, vol. 44(1), pages 27-55, October.
    Full references (including those not matched with items on IDEAS)

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