论文标题
在一系列增加的域及其应用的序列上控制半线性热方程的成本的统一成本
A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application
论文作者
论文摘要
在本文中，我们首先证明了统一的上限对半线性热方程的成本均匀，全球Lipschitz非线性在一系列增加的域上，在整个欧几里得空间r^n中，对控件的作用。作为应用程序，我们在r^n中显示了该半线性热方程的准确无效可控性。这里的主要新颖性是，对于所考虑的域的大小，可以统一地制造出大型但有限域中这种方程式的零控制成本的上限。当一个人使用合适的近似参数来得出R^n中半线性热方程的全局无效可控性时，后者至关重要。这使我们能够克服众所周知的问题，即在通常无界域中非线性PDE的无可控性研究中缺乏紧凑性嵌入。
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 exactly 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.
