On the controllability of the fifth order Korteweg-de Vries equation

This class of equations was introduced by Kichenassamy and Olver [16]. It contains in particular the Kawahara equation [14] introduced to model magneto-acoustic waves, the various models derived by Olver [18] for the unidirectional propagation of waves in shallow water when the third order term appearing in the Korteweg-de Vries equation is small, and many other models. See [16] for a discussion of them. In this paper, we are interested in studying this equation in a bounded domain. We will both consider the Cauchy problem with boundary conditions and the boundary controllability problem. Note that there is an important literature concerning the Cauchy problem in the real line, see for instance [6, 9, 16, 15, 17, 20] and references therein. But for what concerns the boundary value problem, the problem was still completely open as far as we know. Note that the initial boundary value problem for the (third-order) Korteweg-de Vries equation has drained much attention (see in particular [1, 4, 5, 10, 13]). The controllability problem was also, up to our knowledge, completely open. The equivalent for the Korteweg-de Vries equation has also known many developments lately [2, 3, 7, 12, 21, 22, 23, 24, 25]. To be more precise, we will consider in the sequel that α > 0: this is not a restriction since it suffices to make the change of variable x′ = 1 − x and to invert the role of the left and right boundaries. The spatial domain will be [0, 1], which is not a restriction either in the present paper, since it will suffice to rescale in space to obtain a result on an interval of arbitrary length. (Note that this is not necessarily the case for the Korteweg-de Vries equation with Neumann boundary control, see [21]). The boundary conditions that we will consider are the following: