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Article Contents

# Error analysis for global minima of semilinear optimal control problems

• * Corresponding author: Michael Hinze
• 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:

• 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

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
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