In this article, we are concerned with the Galerkin spectral approximation of an $H^1$-norm state-constrained optimal control problem governed by a fourth-order partial differential equation (PDE). In the investigations, the optimality conditions for both the original control problem and its spectral approximation problem are, respectively, established, which are helpful to analyze the solution property of the spectral approximation problem. A priori and a posteriori error estimates of the spectral approximation problem are, therewith, built in detail. Finally, two numerical examples are carried out to validate the theoretical analysis.

中文翻译：

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具有H ^{1 -}范数状态约束的四阶最优控制问题的伽辽金谱方法

在本文中，我们关注一个的伽辽金谱近似 $H^1$由四阶偏微分方程 (PDE) 控制的范数状态约束最优控制问题。研究中分别建立了原控制问题及其谱逼近问题的最优性条件，有助于分析谱逼近问题的解性质。谱近似问题的先验和后验误差估计随之被详细建立。最后，通过两个数值算例验证了理论分析。