A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application
A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application
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DOI：
10.1051/cocv/2022001
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发表时间：
2020-11
期刊：
影响因子：
--
通讯作者：
Lijuan Wang;Can Zhang
中科院分区：
文献类型：
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作者：
Lijuan Wang;Can Zhang
本文首先证明了一类具有全局Lipschitz非线性项的半线性热方程在一系列增域上的零控制代价的一个一致上界，其中控制作用在分布于整个欧氏空间RN的等分布集上.作为一个应用，我们证明了这个半线性热方程的精确零能控性。这里的主要新奇之处是，这种方程在大，但有界域的零控制的成本上界可以统一考虑域的大小。后者是至关重要的，当一个人使用一个合适的近似参数，以获得整体零能控性的半线性热方程在RN。这使我们能够克服众所周知的问题，缺乏紧凑性嵌入所产生的零可控性的非线性偏微分方程在一般无界域的研究。
In this paper, we first prove a uniform upper bound on costs of null controls for semilinear heat equations with globally Lipschitz nonlinearity on a sequence of increasing domains, where the controls are acted on an equidistributed set that spreads out in the whole Euclidean space R N . As an application, we then show the exact null-controllability for this semilinear heat equation in R N . The main novelty here is that the upper bound on costs of null controls for such kind of equations in large but bounded domains can be made uniformly with respect to the sizes of domains under consideration. The latter is crucial when one uses a suitable approximation argument to derive the global null-controllability for the semilinear heat equation in R N . This allows us to overcome the well-known problem of the lack of compactness embedding arising in the study of null-controllability for nonlinear PDEs in generally unbounded domains.