We consider the Kawahara equation, a fifth order Korteweg–de Vries type equation, posed on a bounded interval. The first result of the article is related to the well-posedness in weighted Sobolev spaces, which one was shown using a general version of the Lax–Milgram Theorem. With respect to the control problems, we will prove two results. First, if the control region is a neighborhood of the right endpoint, an exact controllability result in weighted Sobolev spaces is established. Lastly, we show that the Kawahara equation is controllable by regions on $L^2$ Sobolev space, the so-called regional controllability, that is, the state function is exact controlled on the left part of the complement of the control region and null controlled on the right part of the complement of the control region.

我们考虑Kawahara方程，它是有界区间上的五阶Korteweg-de Vries型方程。这篇文章的第一个结果与加权Sobolev空间中的适定性有关，这是使用Lax–Milgram定理的一般形式显示的。关于控制问题，我们将证明两个结果。首先，如果控制区域在右端点的附近，则在加权Sobolev空间中建立精确的可控制性结果。最后，我们证明了川原方程受区域上的控制${\mathrm{大号}}^2$Sobolev空间，即所谓的区域可控性，即状态函数在控制区域的补码的左侧精确控制，而在控制区域的补码的右侧精确控制为零。