*(English)*Zbl 0862.93035

The paper studies the exact controllability and stabilizability problem of the KdV equation:

with periodic boundary conditions: ${\partial}_{x}^{k}u(0,t)={\partial}_{x}^{k}u(2\pi ,t)$, $k=0,1,2$, where $f$ denotes a distributed control input such that ${\int}_{0}^{2\pi}fdx=0$. The exact controllability problem with finite time $T$ is sought first for the linear equation: ${\partial}_{t}u+{\partial}_{x}^{3}u=f$ within the framework of the moment problem: It is solved by introducing an associated Riesz basis (eigenfunctions) and the dual Riesz basis. Then the problem for the original KdV equation is solved by interpreting the term $u{\partial}_{x}u$ as a control via a Fredholm operator. As to the stabilizability problem, the control $f$ is chosen as a feedback of the state $u$ which reduces ${\int}_{0}^{2\pi}{u}^{2}dx$ monotonically. By establishing a discrete decay inequality first, an exponential decay estimate is finally obtained.

##### MSC:

93C20 | Control systems governed by PDE |

35K60 | Nonlinear initial value problems for linear parabolic equations |

93D15 | Stabilization of systems by feedback |

35Q53 | KdV-like (Korteweg-de Vries) equations |