论文标题
通过有限元法对正则非凸的最佳控制问题的数值近似
Numerical approximation of regularized non-convex elliptic optimal control problems by the finite element method
论文作者
论文摘要
我们调查了椭圆形最佳控制问题的数值近似，该问题涉及成本函数中$ l^q $ quasinorm惩罚（$ q \ in（0,1）$）的非convex局部正规化。我们的方法基于\ emph {convex差异}函数公式，该功能得出了一阶必要的最佳条件，可以将其视为辅助凸的最佳系统$ l^1 $ l^1 $ penalizate-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penalizatization-penale contemation。 我们考虑对控件的分段构恒定有限元近似，而状态方程是使用分段线性基函数近似的。然后，为提出的近似值获得收敛结果。在最佳控制支持边界的某些条件下，我们推断出$ h^\ frac12 $收敛率的订单，其中$ h $是关联的离散参数。我们通过数值实验说明了我们的理论发现，这些实验显示了数值近似的收敛行为
We investigate the numerical approximation of an elliptic optimal control problem which involves a nonconvex local regularization of the $L^q$-quasinorm penalization (with $q\in(0,1)$) in the cost function. Our approach is based on the \emph{difference-of-convex} function formulation, which leads to first-order necessary optimality conditions, which can be regarded as the optimality system of an auxiliar convex $L^1$-penalized optimal control problem. We consider piecewise-constant finite element approximation for the controls, whereas the state equation is approximated using piecewise-linear basis functions. Then, convergence results are obtained for the proposed approximation. Under certain conditions on the support's boundary of the optimal control, we deduce an order of $h^\frac12$ approximation rate of convergence where $h$ is the associated discretization parameter. We illustrate our theoretical findings with numerical experiments that show the convergence behavior of the numerical approximation
