In recently proposed stabilization techniques for parabolic equations, a crucial role is played by a suitable sequence of oblique projections in Hilbert spaces, onto the linear span of a suitable set of |$M$| actuators, and along the subspace orthogonal to the space spanned by ‘the’ first |$M$| eigenfunctions of the Laplacian operator. This new approach uses an explicit feedback law, which is stabilizing provided that the sequence of operator norms of such oblique projections remains bounded. The main result of the paper is the proof that, for suitable explicitly given sequences of sets of actuators, the operator norm of the corresponding oblique projections remains bounded. In the final part of the paper we provide numerical results, showing the performance of the explicit feedback control for both Dirichlet and Neumann homogeneous boundary conditions.

中文翻译：

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一维线性非自治抛物方程通过斜投影的显式反馈镇定

在最近提出的抛物线方程的稳定化技术中，希尔伯特空间中合适的倾斜投影序列在合适的| $ M $ |线性范围上起着至关重要的作用。致动器，并且沿着与“ the”第一个| $ M $ |跨越的空间正交的子空间拉普拉斯算子的本征函数。这种新方法使用了显式反馈定律，只要该斜投影的算子范数序列的序列保持有界即可稳定。本文的主要结果是证明，对于合适的显式给定的一组执行器序列，相应的倾斜投影的算子范数仍然是有界的。在本文的最后部分，我们提供了数值结果，显示了在Dirichlet和Neumann齐次边界条件下显式反馈控制的性能。