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Zbl 0862.93035
Russell, David L.; Zhang, Bing-Yu
Exact controllability and stabilizability of the Korteweg-de Vries equation. (English)
[J] Trans. Am. Math. Soc. 348, No.9, 3643-3672 (1996). ISSN 0002-9947; ISSN 1088-6850/e

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.
[T.Nambu (Kobe)]

MSC 2000:: *93C20 Control systems governed by PDE
35K60 (Nonlinear) BVP for (non)linear parabolic equations
93D15 Stabilization of systems by feedback
35Q53 KdV-like equations

Keywords: exact controllability; stabilizability; KdV equation; moment problem; Riesz basis; feedback

Cited in: Zbl 1213.93015

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Zentralblatt MATH Berlin [Germany]