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CONTROL OF A BOUSSINESQ SYSTEM OF KDV-KDV TYPE ON A BOUNDED INTERVAL

机译：有界区间上KDV-KDV型BOUSSINESQ系统的控制。

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摘要

We consider a Boussinesq system of KdV-KdV type introduced by J.L. Bona, M. Chen and J.-C. Saut as a model for the motion of small amplitude long waves on the surface of an ideal fluid. This system of two equations can describe the propagation of waves in both directions, while the single KdV equation is limited to unidirectional waves. We are concerned here with the exact controllability of the Boussinesq system by using some boundary controls. By reducing the controllability problem to a spectral problem which is solved by using the Paley-Wiener method introduced by the third author for KdV, we determine explicitly all the critical lengths for which the exact controllability fails for the linearized system, and give a complete picture of the controllability results with one or two boundary controls of Dirichlet or Neumann type. The extension of the exact controllability to the full Boussinesq system is derived in the energy space in the case of a control of Neumann type. It is obtained by incorporating a boundary feedback in the control in order to ensure a global Kato smoothing effect.
机译：我们考虑由J.L. Bona，M。Chen和J.-C引入的KdV-KdV类型的Boussinesq系统。将Saut作为理想流体表面上小振幅长波运动的模型。这个由两个方程组成的系统可以描述波在两个方向上的传播，而单个KdV方程仅限于单向波。在这里，我们通过使用一些边界控制来关注Boussinesq系统的确切可控性。通过使用第三作者针对KdV引入的Paley-Wiener方法将可控性问题简化为频谱问题，我们明确确定了线性化系统无法精确控制的所有临界长度，并给出了完整的图片一或两个Dirichlet或Neumann型边界控制的可控制性结果。对于Neumann型控制，在能量空间中可以得到精确的可控性扩展到整个Boussinesq系统。通过将边界反馈合并到控件中来确保全局Kato平滑效果。

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  • 来源
    《ESAIM》 |2019年第1期|58.1-58.55|共55页
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  • 作者单位

    Univ Fed Pernambuco UFPE Dept Matemat BR-50740545 Recife PE Brazil;

    Univ Fed Rio de Janeiro Inst Matemat Ilha Fundao CP 68530 BR-21941909 Rio De Janeiro RJ Brazil;

    PSL Res Univ MINES ParisTech CAS 60 Blvd St Michel F-75272 Paris 06 France|PSL Res Univ MINES ParisTech Ctr Robot CAOR 60 Blvd St Michel F-75272 Paris 06 France;

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  • 正文语种 eng
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  • 关键词

    Boussinesq system; KdV-KdV system; exact controllability; stabilization;

    机译：Boussinesq系统;KdV-KdV系统;精确的可控性;稳定化;

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