The boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω are considered. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ωh (typically polygonal) is required. Here, we do not consider the numerical approximation of the control problems. Instead of it, we formulate the corresponding infinite dimensional control problems in Ωh and we study the influence of the replacement of Ω by Ωh on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = δΩ with those defined on Γh = δΩh and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates. The results for convex domains are given in [1], the results for nonconvex domains are included in a work in progress.
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机译：考虑与在弯曲域Ω中定义的半线性椭圆方程相关的边界控制问题。分析了Dirichlet和Neumann的情况。为了对这些问题进行数值分析，需要通过适当的域Ω h （通常是多边形）来近似Ω。在这里，我们不考虑控制问题的数值近似。取而代之的是，我们在Ω h 中制定了相应的无穷维控制问题，并研究了用Ω h 替换Ω对控制问题解的影响。我们的目标是将Γ=δΩ定义的最优控制与Γ h =δΩ h 定义的最优控制进行比较，并得出一些误差估计。对于这种估计，需要使用方便的边界参数化。凸域的结果在[1]中给出，非凸域的结果包括在进行中的工作中。
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