In this paper, an optimal control problem governed by elliptic equations with $L^2$‑norm constraint for state variable is developed. Firstly, the optimality conditions of the optimal control problem are derived, and the optimal control problem is approximated by the Galerkin spectral methods. Similarly, the optimality conditions of the discrete problem are also obtained. Then, some important lemmas are proved to obtain a priori error estimates of the coupled state and control approximation rigorously. Moreover, a posteriori error estimates are also established for the optimal control problem carefully. Finally, based on the projection gradient algorithm, some numerical experiments are presented to confirm our analytical findings. It is proved that the exponential convergence rate can be achieved.

中文翻译：

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具有$L^2$-范数状态约束的椭圆最优控制问题的伽辽金谱方法

在本文中，提出了一个由状态变量$L^2$-范数约束的椭圆方程控制的最优控制问题。首先推导了最优控制问题的最优性条件，并利用伽辽金谱方法对最优控制问题进行了逼近。同样，也得到了离散问题的最优性条件。然后，证明了一些重要的引理来严格地获得耦合状态和控制近似的先验误差估计。此外，后验还仔细地为最优控制问题建立了误差估计。最后，基于投影梯度算法，给出了一些数值实验来证实我们的分析结果。证明可以达到指数收敛速度。