Stabilization of the Korteweg-de Vries equation with localized damping
Authors:
G. Perla Menzala, C. F. Vasconcellos and E. Zuazua
Journal:
Quart. Appl. Math. 60 (2002), 111-129
MSC:
Primary 35Q53; Secondary 35B35, 35B40
DOI:
https://doi.org/10.1090/qam/1878262
MathSciNet review:
MR1878262
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Abstract: We study the stabilization of solutions of the Korteweg-de Vries (KdV) equation in a bounded interval under the effect of a localized damping mechanism. Using multiplier techniques we deduce the exponential decay in time of the solutions of the underlying linear equation. A locally uniform stabilization result of the solutions of the nonlinear KdV model is also proved. The proof combines compactness arguments, the smoothing effect of the KdV equation on the line and unique continuation results.
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J. Bona and R. Winter, The Korteweg-de Vries equation posed in a quarter-plane, SIAM J. Math. Anal. 14, 2056–1106 (1983)
Th. Cazenave and A. Haraux, Introduction aux problèmes d’évolution semi-linéaires, Mathématiques et Applications, Soc. Math. Appl. et Ind., Ellipses, France, 1990
C. M. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29, 241–271 (1968)
A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Differential Equations 59, 145–154 (1985)
A. E. Ingham, Some trigonometrical inequalities with application to the theory of series, Math. Z. 31, 367–379 (1936)
T. Kato, On the Cauchy problem for the (generalized) Korteweg-de Vries equation, Stud. Appl. Math., Adv. in Math. Suppl. Stud. 8, 93–128 (1983)
V. Komornik, Exact controllability and stabilization, the multiplier method, Research in Applied Mathematics 36, John Wiley and Sons, Masson, 1994
V. Komornik, D. L. Russell, and B. Y. Zhang, Stabilisation de l’équation de Korteweg-de Vries, C. R. Acad. Sci. Paris, Série I Math. 312, 841–843 (1991)
S. N. Kruzhkov and A. V. Faminskii, Generalized solutions of the Cauchy problem for the Korteweg-de Vries equation, Math. URSS Sbornik 38, 391–421 (1984)
J. L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1, Contrôlabilité exacte, Collection de Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988
J. L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Tome 1, Dunod, Paris, 1968
S. Micu and E. Zuazua, Boundary controllability of a linear hybrid system arising in the control of noise, SIAM J. Control Optim. 35, 1614–1638 (1997)
R. M. Miura, The Korteweg-de Vries equation, A survey of results, SIAM Review 18, 412–459 (1976)
M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann. 305, 403–417 (1996)
L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain, ESAIM, Control Optimization and Calculus of Variations, vol. 2, 1997, pp. 33–55 (electronic)
G. Perla-Menzala and E. Zuazua, Decay rates for the von Kármán system of thermoelastic plates, Integral and Differential Equations 11, 755–770 (1998) (C. R. Acad. Sci. Paris 324, 49–54 (1997))
A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, J. Math. Pures Appl. 71, 455–467 (1992)
D. L. Russell and B. Y. Zhang, Smoothing and decay properties of solutions of the Korteweg-de Vries equation on a periodic domain with point dissipation, J. Math. Anal. Appl. 190, 449–488 (1995)
D. L. Russell and B. Y. Zhang, Controllability and stabilizability of the third order linear dispersion equation on a periodic domain, SIAM J. Control. Optim. 31, 659–676 (1993)
D. L. Russell and B. Y. Zhang, Exact controllability and stabilization of the Korteweg-de Vries equation, Trans. Amer. Math. Soc. 348, 3643–3672 (1996)
J. C. Saut and B. Scheurer, Unique continuation for some evolution equations, J. Differential Equations 66, 118–139 (1987)
J. Simon, Compact sets in the space ${L^{p}}\left ( 0, T; B \right )$, Annali di Matematica Pura ed Applicata (IV), vol. CXLVI, 1987, pp. 65–96
M. Slemrod, Weak asymptotic decay via a “Relaxed Invariance Principle” for a wave equation with nonlinear, nonmonotone damping, Proc. Royal Soc. Edinburgh 113, 87–97 (1989)
R. Temam, Sur un problème non linéaire, J. Math. Pures et Appl. 38, 157–172 (1969)
R. Teman, Navier-Stokes Equations, Studies in Mathematics and its Applications 2, North-Holland, 1977
B. Y. Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J. Math. Anal. 23, 55–71 (1991)
B. Y. Zhang, Exact boundary controllability of the Korteweg-de Vries equation, SIAM J. Control Optim. 37, 548–565 (1999)
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations 15 (2), 205–235 (1990)
E. Zuazua, Contrôlabilité exacte en un temps arbitrairement petit de quelques modèles de plaques, Appendix 1 in [10], Tome I, pp. 465–491
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