In this paper we consider the problem of controlling pointwise, by means of a time dependent Dirac measure supported by a given point, a coupled system of two Korteweg-de Vries equations on the unit circle. More precisely, by means of spectral analysis and Fourier expansion we prove, under general assumptions on the physical parameters of the system, a pointwise observability inequality which leads to the pointwise controllability by using two control functions. In addition, with a uniqueness property proved for the linearized system without control, we are also able to show pointwise controllability when only one control function acts internally. In both cases we can find, under some assumptions on the coefficients of the system, the sharp time of the controllability.

中文翻译：

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线性化Gear-Grimshaw系统的逐点控制

在本文中，我们考虑通过在给定点支持下的时间相关Dirac测度，在单位圆上的两个Korteweg-de Vries方程的耦合系统来逐点控制问题。更准确地说，通过频谱分析和傅立叶展开，我们在系统物理参数的一般假设下证明了点向可观性不等式，该不等式通过使用两个控制函数导致点向可控性。此外，通过对无需控制的线性化系统证明了其独特性，当内部只有一个控制功能起作用时，我们也能够显示逐点可控性。在这两种情况下，在对系统系数的一些假设下，我们都可以发现可控性的急剧时间。