Controllability cost of conservative systems: resolvent condition and transmutation
Controllability cost of conservative systems: resolvent condition and transmutation
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DOI：
10.1016/j.jfa.2004.02.001
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发表时间：
2004-02
影响因子：
1.7
通讯作者：
Luc Miller
中科院分区：
文献类型：
--
作者：
Luc Miller
本文研究了具有无界控制算子的Hilbert空间上酉群的精确能控性。它提供了一个不涉及时间的充要条件，它混合了一个预解估计和一个可观测性不等式。通过对相应的二阶保守系统的控制在某一时刻的变形，证明了当T趋于0时，酉群在时间T上的控制代价最多像EXP(αL2/T)那样增长.在薛定谔方程快速控制费用的应用中，L是几何光学中不与控制区域相交的最长射线的长度。文中还给出了能观测性可解估计，并指出系统的可控性代价不因保守系统的张量积而改变。
This article concerns the exact controllability of unitary groups on Hilbert spaces with unbounded control operator. It provides a necessary and sufficient condition not involving time which blends a resolvent estimate and an observability inequality. By the transmutation of controls in some time L for the corresponding second-order conservative system, it is proved that the cost of controls in time T for the unitary group grows at most like exp(αL2/T) as T tends to 0. In the application to the cost of fast controls for the Schrödinger equation, L is the length of the longest ray of geometric optics which does not intersect the control region. This article also provides observability resolvent estimates implying fast smoothing effect controllability at low cost, and underscores that the controllability cost of a system is not changed by taking its tensor product with a conservative system.