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Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation

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Abstract

In this article, we continue our study of a system described by a class of initial boundary value problem (IBVP) of the Korteweg-de Vries (KdV) equation and the KdV Burgers (KdVB) equation posed on a finite interval with nonhomogeneous boundary conditions. While the system is known to be locally well-posed (Kramer et al. arXiv:1012.1057, [2010]; Rivas et al. in Math. Control Relat. Fields 1:61–81, [2011]) and its small amplitude solutions are known to exist globally, it is not clear whether its large amplitude solutions would blow up in finite time or not. This problem is addressed in this article from control theory point of view: look for some appropriate feedback control laws (with boundary value functions as control inputs) to ensure that the finite time blow-up phenomena would never occur. In this article, a simple, but nonlinear, feedback control law is proposed and the resulting closed-loop system is shown not only to be globally well-posed, but also to be locally exponentially stable for the KdV equation and globally exponentially stable for the KdVB equation.

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Notes

  1. The interested readers are refereed to a nice article of de Jager [25] for the origin of the KdV equation.

  2. The reader is referred to [28] for the precise definition of s-compatibility for the IBVP (1.2). One of the sufficient conditions for ϕ,h 1,h 2,h 3 to be s-compatible is \(\phi\in H^{s}_{0} (0,L)\) and

    $$h_1\in H^{\frac{s+1}{3}}_0 (0,T], \quad h_2\in H^{\frac{s}{3}}_0 (0,T],\quad h_3\in H^{\frac{s-1}{3}}_0 (0,T].$$

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Acknowledgements

Chaohua Jia was partially supported by State Scholarship Fund of China (No. 2010602510) from China Scholarship Council (CSC), the National Natural Science Foundation of China (No.10626002) and the Fundamental Research Funds YWF-10-01-A15 for the Central Universities. Bing-Yu Zhang was partially supported by a grant from the Simons Foundation (#201615 to Bingyu Zhang), the Taft Memorial Fund at the University of Cincinnati and the Chunhui program (State Education Ministry of China) under grant Z007-1-61006. This work was conducted while Chaohua Jia was visiting Department of Mathematical Science, University of Cincinnati from February 2011 to February 2012 under the State Scholarship Fund of China. He would like to thank the University of Cincinnati for its hospitality.

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Authors and Affiliations

  1. School of Mathematics and Systems Science, Beihang University, Beijing, 100191, China

    Chaohua Jia

  2. LMIB of the Ministry of Education, Beijing, 100191, China

    Chaohua Jia

  3. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221, USA

    Bing-Yu Zhang

  4. Yangtz Center of Mathematics, Sichuan University, Chengdu, China

    Bing-Yu Zhang

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  1. Chaohua Jia
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Correspondence to Bing-Yu Zhang.

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The article is dedicated to Professor Goong Chen for his 60th birthday.

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Jia, C., Zhang, BY. Boundary Stabilization of the Korteweg-de Vries Equation and the Korteweg-de Vries-Burgers Equation. Acta Appl Math 118, 25–47 (2012). https://doi.org/10.1007/s10440-012-9676-4

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