In this work, we derive an a priori error estimate of order for the finite element approximation of a sparse optimal control problem governed by an elliptic equation, which is controlled in a finite dimensional space. Furthermore, box‐constrains on the control are considered and finitely many pointwise state‐constrains are imposed on specific points in the domain. With this choice for the control space, the achieved order of approximation for the optimal control is optimal, in the sense that the order of the error for the optimal control is of the same order of the approximation for the state equation.

中文翻译：

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具有点状态约束和有限维控制空间的椭圆方程最优稀疏控制的FEM逼近的误差估计

在这项工作中，我们推导了由椭圆方程控制的稀疏最优控制问题的有限元近似的阶次先验误差估计，该方程在有限维空间中控制。此外，考虑了控件上的盒约束，并且在域中的特定点上强制施加了许多点状状态约束。在控制空间的这种选择下，在最佳控制的误差阶数与状态方程的近似阶数相同的意义上，最佳控制的近似逼近级是最佳的。