Sharp estimates and homogenization of the control cost of the heat equation on large domains
Sharp estimates and homogenization of the control cost of the heat equation on large domains
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DOI：
10.1051/cocv/2019058
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发表时间：
2018-10
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影响因子：
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通讯作者：
Ivica Naki'c;Matthias Täufer;Martin Tautenhahn;I. Veselić
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文献类型：
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作者：
Ivica Naki'c;Matthias Täufer;Martin Tautenhahn;I. Veselić
我们证明了新的边界上的控制成本的抽象热方程，假设光谱不等式或光谱投影仪的不确定性关系。特别是，我们定量地指定控制成本的上限取决于谱不等式中的常数。然后将其应用于由薛定谔半群建模的有界和无界域上的热流。这意味着允许放热发生器包含势项。可观测性/控制集被假设为服从等分布或厚度条件，这取决于上下文。互补的下界和例子表明，我们的控制成本估计是尖锐的，在某些渐近制度。其中之一被称为均匀化制度，对应于控制集在整个域中变得越来越均匀分布而其密度保持恒定的情况。
We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation where the control set becomes more and more evenly distributed throughout the domain while its density remains constant.