Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations
Discrete Carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations
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DOI：
10.1016/j.matpur.2009.11.003
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发表时间：
2010-03-01
影响因子：
2.3
通讯作者：
Le Rousseau, Jerome
中科院分区：
文献类型：
--
作者：
Boyer, Franck;Hubert, Florence;Le Rousseau, Jerome
我们得到了一个半离散的二维椭圆整体Carleman估计，其中通常的大参数连接到一维离散步长。我们处理的离散化是一些家庭的平滑变化的网格。作为Carleman估计的结果，我们得到了一个由G. Lebeau和L. Robbiano，在一维离散椭圆算子的情况下。这里，这个不等式涉及离散谱的较低部分。然而，我们处理的特征值/特征函数的范围是准最优的，并且代表离散谱的恒定部分。对于相关的抛物型问题，我们然后得到一个统一的零可控性的结果，这较低的部分的频谱。此外，我们构造的控制函数，最终状态的L-2-范数收敛到零超代数的离散化的步长为零。然后推导出一个松弛的可观测性估计。(C)2009年Elsevier Masson SAS。All rights reserved.
We derive a semi-discrete two-dimensional elliptic global Carleman estimate, in which the usual large parameter is connected to the one-dimensional discretization step-size. The discretizations we address are some families of smoothly varying meshes. As a consequence of the Carleman estimate, we derive a partial spectral inequality of the form of that proven by G. Lebeau and L. Robbiano, in the case of a discrete elliptic operator in one dimension. Here, this inequality concerns the lower part of the discrete spectrum. The range of eigenvalues/eigenfunctions we treat is however quasi-optimal and represents a constant portion of the discrete spectrum. For the associated parabolic problem, we then obtain a uniform null controllability result for this lower part of the spectrum. Moreover, with the control function that we construct, the L-2-norm of the final state converges to zero super-algebraically as the step-size of the discretization goes to zero. A relaxed observability estimate is then deduced. (C) 2009 Elsevier Masson SAS. All rights reserved.