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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:

t u+u x u+ x 3 u=f,0x2π,t0

with periodic boundary conditions: x k u(0,t)= x k u(2π,t), k=0,1,2, where f denotes a distributed control input such that 0 2π fdx=0. The exact controllability problem with finite time T is sought first for the linear equation: t u+ 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 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 0 2π u 2 dx monotonically. By establishing a discrete decay inequality first, an exponential decay estimate is finally obtained.

Reviewer: T.Nambu (Kobe)

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