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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: $$\partial_tu+u \partial_xu+\partial^3_xu=f,\quad 0\le x\le 2\pi,\quad t\ge 0$$ with periodic boundary conditions: $\partial^k_xu(0,t)= \partial^k_xu(2\pi,t)$, $k=0,1,2$, where $f$ denotes a distributed control input such that $\int^{2\pi}_0 f dx=0$. The exact controllability problem with finite time $T$ is sought first for the linear equation: $\partial_tu+\partial^3_xu=f$ 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 $u\partial_xu$ 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 $\int^{2\pi}_0 u^2dx$ monotonically. By establishing a discrete decay inequality first, an exponential decay estimate is finally obtained.
Reviewer: T.Nambu (Kobe)

93C20Control systems governed by PDE
35K60Nonlinear initial value problems for linear parabolic equations
93D15Stabilization of systems by feedback
35Q53KdV-like (Korteweg-de Vries) equations
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