Exact Boundary Controllability of Fifth-Order KdV Equation Posed on the Periodic Domain

Abstract

In this paper, we show by Hilbert Uniqueness Method that the boundary value problem of fifth-order KdV equation\begin{align*}\begin{cases}y_{t}-y_{5 x} =0, \quad(x, t) \in(0,2 \pi) \times(0, T),\\y(t, 2 \pi)-y(t, 0) =h_{0}(t),\\y_{x}(t, 2 \pi)-y_{x}(t, 0) =h_{1}(t),\\y_{2 x}(t, 2 \pi)-y_{2 x}(t, 0) =h_{2}(t),\\y_{3 x}(t, 2 \pi)-y_{3 x}(t, 0) =h_{3}(t),\\y_{4 x}(t, 2 \pi)-y_{4 x}(t, 0) =h_{4}(t),\end{cases}\end{align*}

(with boundary data as control inputs) is exact controllability.