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Stabilization of the Korteweg-de Vries Equation on a Periodic Domain

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Control and Optimal Design of Distributed Parameter Systems

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 70))

Abstract

We study solutions of the Korteweg-de Vries (KdV) equations

$$ {u_t} + u{u_x} + {u_{xxx}} = f $$

and

$$ {u_t} + u{u_x} + {u_{xxx}} = 0 $$

for t ≥ 0 and 0 ≤ x ≤ 1 where the subscripts denote partial derivatives. hi the first case, periodic boundary conditions are imposed at 0 and 1, and the distributed control f is assumed to be generated by a linear feedback control law conserving the “volume” or “mass” ∫ 10 u(x, t)dx which monotonically reduces the “energy” ∫ 10 u(x, t)2 dx. For the second equation a feedback boundary control is applied having the same properties. In both cases we obtain uniform exponential decay of the solutions to a constant state.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 24061-0123, USA

    David L. Russell

  2. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio, 45221, USA

    Bing-Yu Zhang

Authors
  1. David L. Russell
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  2. Bing-Yu Zhang
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Editor information

Editors and Affiliations

  1. Department of Mathematics, Georgetown University, 20057, Washington, DC, USA

    John E. Lagnese

  2. Department of Mathematics, Virginia Polytechnic and State University, 24061-0123, Blacksburg, VA, USA

    David L. Russell

  3. Department of Mathematics, University of Oklahoma, 601 Elm Avenue, Room 423, 73019-0315, Norman, OK, USA

    Luther W. White

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© 1995 Springer-Verlag New York, Inc.

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Russell, D.L., Zhang, BY. (1995). Stabilization of the Korteweg-de Vries Equation on a Periodic Domain. In: Lagnese, J.E., Russell, D.L., White, L.W. (eds) Control and Optimal Design of Distributed Parameter Systems. The IMA Volumes in Mathematics and its Applications, vol 70. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8460-1_9

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  • DOI: https://doi.org/10.1007/978-1-4613-8460-1_9

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8462-5

  • Online ISBN: 978-1-4613-8460-1

  • eBook Packages: Springer Book Archive

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