Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem
Convergence of a Finite Element Approximation to a State-Constrained Elliptic Control Problem
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DOI：
10.1137/060652361
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发表时间：
2007-08
期刊：
影响因子：
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通讯作者：
K. Deckelnick;M. Hinze
中科院分区：
文献类型：
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作者：
K. Deckelnick;M. Hinze
本文考虑一类带逐点状态约束的椭圆型最优控制问题。成本泛函近似的泛函序列，这是通过离散的状态方程的帮助下，线性有限元和强制执行的三角形的节点中的状态约束。相应的最小值收敛在$L^2$的精确控制的离散化参数趋于零。此外，在两个和三个空间维的控制和状态的误差界。最后，我们提出的数值例子，证实了我们的分析结果。
We consider an elliptic optimal control problem with pointwise state constraints. The cost functional is approximated by a sequence of functionals which are obtained by discretizing the state equation with the help of linear finite elements and enforcing the state constraints in the nodes of the triangulation. The corresponding minima are shown to converge in $L^2$ to the exact control as the discretization parameter tends to zero. Furthermore, error bounds for the control and the state are obtained in both two and three space dimensions. Finally, we present numerical examples which confirm our analytical findings.