arXiv 1902.058932019-02math.OC

一维椭圆型最优控制的BV函数有限元误差估计

Finite element error estimates for one-dimensional elliptic optimal control by BV functions

Dominik Hafemeyer, Florian Mannel, Ira Neitzel, Boris Vexler

摘要：

考虑一维椭圆型方程的最优控制问题，该方程涉及有界变差的单变量函数作为控制。对于状态方程的离散化，我们使用线性有限元和控制离散化，我们分析了两种策略。首先，我们使用变分离散化的控制和$L^2$ -和$L^\infty$ -错误的状态和伴随状态的顺序为${\mathcal O}(h^2)$和$L^1$ -错误的控制行为像${\mathcal O}(h^2)$，太。这些结果依赖于一个结构性的假设，这意味着原来的问题的最优控制是分段常数和伴随状态有非零的第一阶导数的控制的跳跃点。如果，第二，分段常数控制离散化，我们得到$L^2$ -误差估计的顺序$\mathcal{O}(h)$的状态和$W^{1,\infty}$ -误差估计的顺序$\mathcal{O}(h)$的伴随状态。在相同的结构假设下，我们推导出控制的$L^1$ -误差估计为$\mathcal{O}(h)$阶。我们讨论了优化算法，并提供了数值结果的离散计划表明，误差估计是最佳的。

英文摘要：

We consider an optimal control problem governed by a one-dimensional elliptic equation that involves univariate functions of bounded variation as controls. For the discretization of the state equation we use linear finite elements and for the control discretization we analyze two strategies. First, we use variational discretization of the control and show that the $L^2$- and $L^\infty$-error for the state and the adjoint state are of order ${\mathcal O}(h^2)$ and that the $L^1$-error of the control behaves like ${\mathcal O}(h^2)$, too. These results rely on a structural assumption that implies that the optimal control of the original problem is piecewise constant and that the adjoint state has nonvanishing first derivative at the jump points of the control. If, second, piecewise constant control discretization is used, we obtain $L^2$-error estimates of order $\mathcal{O}(h)$ for the state and $W^{1,\infty}$-error estimates of order $\mathcal{O}(h)$ for the adjoint state. Under the same structural assumption as before we derive an $L^1$-error estimate of order $\mathcal{O}(h)$ for the control. We discuss optimization algorithms and provide numerical results for both discretization schemes indicating that the error estimates are optimal.

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英文摘要：

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