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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: \[ \partial_tu+u \partial_xu+\partial^3_xu=f,\quad 0\leq x\leq 2\pi,\quad t\geq 0 \] 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)

MSC:
93C20 Control/observation systems governed by partial differential equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
93D15 Stabilization of systems by feedback
35Q53 KdV equations (Korteweg-de Vries equations)
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