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Boundary Stabilization of the Korteweg-De Vries Equation

  • Bing-Yu Zhang
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 118)

Abstract

Consider herein is the initial-boundary value problem of the KdV equation posed on the bounded interval (0, 1):
$$ (*) \left\{ {\begin{array}{*{20}{c}} {{{u}_{t}} + u{{u}_{x}} + {{u}_{{xxx, }}}u(x,0) = \emptyset (x)} \\ {u(0,t) = 0, u(1,t) = 0, {{u}_{x}}(1,t) = \alpha {{u}_{x}}(0,t)} \\ \end{array} } \right. $$
where |α| < 1. It is shown that (i) the system (*) is globally well-posed in the space
$$ H_{\alpha }^{3} = \{ \emptyset \in {{H}^{3}}(0,1), \emptyset (0,t) = 0, \emptyset (1,t) = \alpha {{\emptyset }_{x}}(0,t)\} $$
and (ii) if α ≠ 0, then the system (*) is locally well-posed in the space H 0 1 (0, 1), but its small amplitude solutions exist globally and decay exponentially to zero as t → ∞.

1991 Mathematics Subject Classification

35Q20 93D15 93C20 93B05 

Key words and phrases

Korteweg-de Vries equation stabilization boundary control 

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Copyright information

© Springer Basel AG 1994

Authors and Affiliations

  • Bing-Yu Zhang
    • 1
  1. 1.Department of Mathematical SciencesUniversity of CincinnatiCincinnatiUSA

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