Exact controllability and stabilizability of the Korteweg-de Vries equation.

*(English)*Zbl 0862.93035The 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) |

##### Keywords:

exact controllability; stabilizability; KdV equation; moment problem; Riesz basis; feedback
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\textit{D. L. Russell} and \textit{B.-Y. Zhang}, Trans. Am. Math. Soc. 348, No. 9, 3643--3672 (1996; Zbl 0862.93035)

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##### References:

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