Mobile Point Controls Versus Locally Distributed Ones for the Controllability of the Semilinear Parabolic Equation
Mobile Point Controls Versus Locally Distributed Ones for the Controllability of the Semilinear Parabolic Equation
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DOI：
10.1137/s0363012999358038
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发表时间：
2001
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影响因子：
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通讯作者：
A. Khapalov
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文献类型：
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作者：
A. Khapalov
现在众所周知，当在有界域中由局部分布控制控制时，具有全局 Lipschitz 非线性项的相当一般的半线性抛物线方程在 L2 (Ω)$ 中既近似又精确地零可控。在本文中，我们打算证明，实际上，在一个空间维度（$\Omega= (0, 1)$）中，通过使用{\em至多}两个移动点控件，并支持在 QT = (0,1) × (0,T) 的任意子域内正确选择的曲线，可以实现完全相同的结果。我们将证明这样的曲线可以用一定的微分不等式来描述，并提供明确的例子。我们还讨论了我们的主要结果对超线性项和多维情况的一些扩展。
It is well known now that a rather general semilinear parabolic equation with globally Lipschitz nonlinear term is both approximately and exactly null-controllable in L2 (\Omega)$, when governed in a bounded domain by the locally distributed controls. In this paper we intend to show that, in fact, in one space dimension ($\Omega= (0, 1)$) the very same results can be achieved by employing {\em at most} two mobile point controls with support on the curves properly selected within an arbitrary subdomain of QT = (0,1) × (0,T). We will show that such curves can be described by a certain differential inequality and the explicit examples are provided. We also discuss some extensions of our main results to the superlinear terms and to the case of several dimensions.