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Error analysis for global minima of semilinear optimal control problems

  • * Corresponding author: Michael Hinze

    * Corresponding author: Michael Hinze
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  • In [2] we consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization, where we provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to the discretization parameter the sequence of discrete solutions converges to a global solution of the corresponding limit problem. With the present work we complement our investigations of [2] in that we prove an error estimate for those discrete global solutions. Numerical experiments confirm our analytical findings.

    Mathematics Subject Classification: Primary: 65K10, 90C26; Secondary: 35Q93.

    Citation:

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  • Figure 1.  Example 1: $\|\bar p_h\|_{L^4}$ and $\eta(\alpha)$ vs. $\alpha$.

    Figure 2.  Example 1: The unique global minimum together with its state and the associated multipliers.

    Figure 3.  Example 1: Errors for the optimal control and its state versus the mesh size.

    Figure 5.  Example 2: The unique global minimum together with its state and the associated multipliers.

    Figure 6.  Example 2: Errors for the optimal control and its state versus the mesh size.

    Table 1.  Example 1: EOC for the optimal control and its state.

    Levels EOC$_{u_{L2}}$ EOC$_{y_{H1}}$ EOC$_{y_{L2}}$ EOC$_{y_{L\infty}}$
    2-3 1.187645 0.833842 1.464788 1.058334
    3-4 1.078183 0.948273 1.708822 1.758387
    4-5 1.027290 0.985352 1.794456 1.657899
    5-6 1.016702 0.997996 1.831198 1.514376
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    Table 2.  Example 2: EOC for the optimal control and its state.

    Levels EOC$_{u_{L2}}$ EOC$_{y_{H1}}$ EOC$_{y_{L2}}$ EOC$_{y_{L\infty}}$
    2-3 1.910131 1.015420 1.686197 1.399920
    3-4 1.983722 1.017850 1.934978 1.876141
    4-5 1.982064 1.005320 1.996849 1.993434
    5-6 1.944839 1.003374 2.035042 2.027269
     | Show Table
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  • [1] A. Ahmad Ali, Optimal Control of Semilinear Elliptic PDEs with State Constraints -Numerical Analysis and Implementation, PhD thesis, Dissertation, Hamburg, Universität Hamburg, 2017.
    [2] A. Ahmad AliK. Deckelnick and M. Hinze, Global minima for semilinear optimal control problems, Computational Optimization and Applications, 65 (2016), 261-288. 
    [3] N. AradaE. Casas and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, 23 (2002), 201-229. 
    [4] E. CasasL2 Estimates for the finite element method for the Dirichlet problem with singular data, Numerische Mathematik, 47 (1985), 627-632. 
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    [13] E. CasasM. Mateos and B. Vexler, New regularity results and improved error estimates for optimal control problems with state constraints, ESAIM. Control, Optimisation and Calculus of Variations, 20 (2014), 803-822. 
    [14] E. Casas and F. Tröltzsch, Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2010), 581-600. 
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