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Exact controllability and stabilizability of the Korteweg-de Vries equation. (English) Zbl 0862.93035

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

${\partial }_{t}u+u{\partial }_{x}u+{\partial }_{x}^{3}u=f,\phantom{\rule{1.em}{0ex}}0\le x\le 2\pi ,\phantom{\rule{1.em}{0ex}}t\ge 0$

with periodic boundary conditions: ${\partial }_{x}^{k}u\left(0,t\right)={\partial }_{x}^{k}u\left(2\pi ,t\right)$, $k=0,1,2$, 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.

Reviewer: T.Nambu (Kobe)

##### 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