Equivalence between Minimal Time and Minimal Norm Control Problems for the Heat Equation
Equivalence between Minimal Time and Minimal Norm Control Problems for the Heat Equation
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DOI：
10.1137/16m1095159
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发表时间：
2016-09
期刊：
影响因子：
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通讯作者：
Shulin Qin;GengshengBB Wang
中科院分区：
文献类型：
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作者：
Shulin Qin;GengshengBB Wang
本文给出了内控热方程的极小时间控制问题与极小范数控制问题的等价性。目标是状态空间中任意固定的有界闭凸集，其内部非空。本研究不同于[G. Wang和E. Zuazua，\textit{关于内部控制热方程的最小时间和最小范数控制的等价性}，SIAM J.控制优化，50（2012），pp. 2938-2958]，其中目标集合是状态空间中的原点。当目标集是原点或以原点为中心的球时，极小范数和极小时间函数是连续的、严格递减的，并且它们是彼此的逆。然而，当目标位于状态空间的其他位置时，最小范数函数可能不再单调，并且最小时间函数的范围可能不连通。这是我们研究的主要困难。我们通过从经典的升日引理中借用一些思想来克服这个困难（例如，参见[E]中引理3.5和图5。M. Stein和R. Shakarchi，\textit{真实的分析：测度理论、积分和希尔伯特空间}，普林斯顿大学出版社，2005]）。
This paper presents the equivalence between minimal time and minimal norm control problems for internally controlled heat equations. The target is an arbitrarily fixed bounded, closed and convex set with a nonempty interior in the state space. This study differs from [G. Wang and E. Zuazua, \textit{On the equivalence of minimal time and minimal norm controls for internally controlled heat equations}, SIAM J. Control Optim., 50 (2012), pp. 2938-2958] where the target set is the origin in the state space. When the target set is the origin or a ball, centered at the origin, the minimal norm and the minimal time functions are continuous and strictly decreasing, and they are inverses of each other. However, when the target is located in other place of the state space, the minimal norm function may be no longer monotonous and the range of the minimal time function may not be connected. These cause the main difficulty in our study. We overcome this difficulty by borrowing some idea from the classical raising sun lemma (see, for instance, Lemma 3.5 and Figure 5 on Pages 121-122 in [E. M. Stein and R. Shakarchi, \textit{Real Analysis: Measure Theory, Integration, and Hilbert Spaces}, Princeton University Press, 2005]).