In this paper, we deal with the multidimensional heat equation governed by a bilinear control. We first investigate the finite-time stability and null controllability properties. Then, we study the global and partial approximate controllability. These results are further applied to establish the exact controllability of the system at hand. Nonnegative controllability results are also discussed. The approaches are based on linear semigroup theory and the controllability of the heat equation with additive controls. An other essential tool that we use here is the classical maximum principle.

中文翻译：

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双线性热方程的有限时间控制

在本文中，我们处理由双线性控制控制的多维热方程。我们首先研究了有限时间稳定性和零可控性。然后，我们研究了全局和部分近似可控性。这些结果将进一步用于建立手头系统的精确可控性。还讨论了非负可控性结果。该方法基于线性半群理论和带有加性控制的热方程的可控性。我们在这里使用的另一个基本工具是经典最大原理。