Local Controllability of Multidimensional Quasi-Linear Parabolic Equations
Local Controllability of Multidimensional Quasi-Linear Parabolic Equations
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DOI：
10.1137/110851808
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发表时间：
2012-07
期刊：
影响因子：
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通讯作者：
X. Liu;Xu Zhang
中科院分区：
文献类型：
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作者：
X. Liu;Xu Zhang
研究了一类具有齐次Dirichlet边界条件和任意位置的内部控制器的多维拟线性抛物方程的局部能控性。与已有的一维结果不同，我们在经典解的框架下分析了系统的能控性。该方法的关键是在具有一定规律性的给定数据的Holder空间中寻找控制函数。为了给出控制函数的代价估计，我们建立了线性抛物型方程的能观测性不等式，并用主成分系数C^1给出了能观测性常数的一个显式估计。后者基于一般线性抛物型方程的一个新的全局Carleman估计，该估计具有独立的意义。
This paper studies the local controllability for a class of multidimensional quasi-linear parabolic equations with homogenous Dirichlet boundary conditions and an arbitrarily located internal controller. Unlike the known result in one spacial dimension, controllability is analyzed in the frame of classical solutions. The key of our approach is to seek a control function in the Holder space for given data with certain regularity. In order to give the cost estimate for the control function, we establish an observability inequality for linear parabolic equations with an explicit estimate on the observability constant in terms of the $C^1$ coefficients of principal parts. The later is based on a new global Carleman estimate for general linear parabolic equations, which has an independent interest.