SIAM Journal on Control and Optimization, Volume 58, Issue 4, Page 1939-1960, January 2020.
In this paper we establish an observability inequality for the heat equation with bounded potentials on the whole space. Roughly speaking, such a kind of inequality says that the total energy of solutions can be controlled by the energy localized in a subdomain, which is equidistributed over the whole space. The proof of this inequality is mainly adapted from the parabolic frequency function method, which plays an important role in proving the unique continuation property for solutions of parabolic equations. As an immediate application, we show that the null controllability holds for the heat equation with bounded potentials on the whole space.

中文翻译：

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在整个空间上具有界势的热方程的可观测性不等式

SIAM控制与优化杂志，第58卷，第4期，第1939-1960页，2020
年1月。在本文中，我们建立了在整个空间上具有有限电势的热方程的可观察性不等式。粗略地说，这种不等式表示解决方案的总能量可以由子域中的能量控制，该子域分布在整个空间中。该不等式的证明主要来自抛物线频率函数法，该方法在证明抛物线方程解的唯一连续性方面起着重要作用。作为立即应用，我们证明了在整个空间上具有有限电势的热方程的零可控性成立。