Control of a Boussinesq system of KdV–KdV type on a bounded interval

Capistrano–Filho, Roberto A.; Pazoto, Ademir F.; Rosier, Lionel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 58.

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

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.

[1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of the waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. | DOI | MR | Zbl

[2] J. Bergh and J. Löfström, Interpolation Spaces: An introduction. Grundlehren der Mathematishen Wissenschaften. Springer-Verlag, Berlin, New York (1976). | DOI | MR | Zbl

[3] J.L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I. Derivation and linear theory. J. Nonlinear Sci. 12 (2002) 283–318. | DOI | MR | Zbl

[4] J.L. Bona, M. Chen and J.-C. Saut, Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. II. The nonlinear theory. Nonlinearity 17 (2004) 925–952. | DOI | MR | Zbl

[5] J. Boussinesq, Théorie générale des mouvements qui sont propagés dans un canal rectangulaire horizontal. C. R. Acad. Sci. Paris 73 (1871) 256–260. | JFM

[6] J. Boussinesq, Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17 (1872) 55–108. | JFM | Numdam | MR

[7] R.A. Capistrano-Filho, A.F. Pazoto, L. Rosier, Internal controllability of the Kortewegde Vries equation on a bounded domain. ESAIM: COCV 21 (2015) 1076–1107. | Numdam | MR | Zbl

[8] E. Cerpa, Exact controllability of a nonlinear Korteweg-de Vries equation on a critical spatial domain. SIAM J. Control Optim. 46 (2007) 877–899. | DOI | MR | Zbl

[9] E. Cerpa and E. Crépeau, Boundary controllability for the nonlinear Korteweg-de Vries equation on any critical domain. Ann. Inst. Henri Poincaré 26 (2009) 457–475. | DOI | Numdam | MR | Zbl

[10] E. Cerpa and E. Crépeau, Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete Contin. Dyn. Syst. Ser. B 11 (2009) 655–668. | MR | Zbl

[11] J. Chu, J.-M. Coron and P. Shang, Asymptotic stability of a nonlinear Korteweg-de Vries equation with critical lengths. J. Differ. Equ. 259 (2015) 4045–4085. | DOI | MR | Zbl

[12] J.-M. Coron, Control and Nonlinearity. In Vol. 136 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2007. | MR | Zbl

[13] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with a critical length. J. Eur. Math. Soc. 6 (2004) 367–398. | DOI | MR | Zbl

[14] P.C.N. Da Silva and C.F. Vasconcellos, On the stabilization and controllability of a third order linear equation. Port. Math. 68 (2011) 279–296. | DOI | MR | Zbl

[15] G.G. Doronin, N. A. Larkin, Stabilization of regular solutions for the Zakharov-Kuznetsov equation posed on bounded rectangles and on a strip. Proc. Edinb. Math. Soc. 58 (2015) 661–682. | DOI | MR | Zbl

[16] G.G. Doronin and F.M. Natali, An example of non-decreasing solution for the KdV equation posed on a bounded interval. C. R. Math. Acad. Sci. Paris 352 (2014) 421–424. | DOI | MR | Zbl

[17] A.L.C. Dos Santos, P.N. Da Silva and C.F. Vasconcellos, Entire functions related to stationary solutions of the Kawahara equation. Electron. J. Differ. Equ. 43 (2016) 13. | MR

[18] O. Glass and S. Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit. Asymptot. Anal. 60 (2008) 61–100. | MR | Zbl

[19] O. Glass and S. Guerrero, Controllability of the Korteweg-de Vries equation from the right Dirichlet boundary condition. Syst. Control Lett. 59 (2010) 390–395. | DOI | MR | Zbl

[20] D.J. Korteweg and G. De Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves. Philos. Mag. 39 (1895) 422–443. | DOI | JFM | MR

[21] C. Laurent, L. Rosier and B.Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain. Commun. Part. Differ. Equ. 35 (2010) 707–744. | DOI | MR | Zbl

[22] F. Linares and A.F. Pazoto, On the exponential decay of the critical generalized Korteweg-de Vries equation with localized damping. Proc. Am. Math. Soc. 135 (2007) 1515–1522. | DOI | MR | Zbl

[23] F. Linares and L. Rosier, Control and Stabilization of the Benjamin-Ono equation on a periodic domain. Trans. AMS 367 (2015) 4595–4626. | DOI | MR | Zbl

[24] J.-L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués – Tome 1. Vol. 8 of Recherches en Mathématiques Appliquées [ Research in Applied Mathematics]. Masson, Paris (1988). | MR | Zbl

[25] J.L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1 of Travaux et Recherches Mathemátiques. Dunod, Paris (1968). | Zbl

[26] S. Micu, J.H. Ortega, L. Rosier and B.-Y. Zhang, Control and stabilization of a family of Boussinesq systems. Discrete Contin. Dyn. Syst. 24 (2009) 273–313. | DOI | MR | Zbl

[27] A.F. Pazoto, Unique continuation and decay for the Korteweg-de Vries equation with localized damping. ESAIM: COCV 11 (2005) 473–486. | Numdam | MR | Zbl

[28] A.F. Pazoto and L. Rosier, Stabilization of a Boussinesq system of KdV–KdV type. Syst. Control Lett. 57 (2008) 595–601. | DOI | MR | Zbl

[29] G. Perla-Menzala, C.F. Vasconcellos and E. Zuazua, Stabilization of the Korteweg-de Vries equation with localized damping. Quart. Appl. Math. 60 (2002) 111–129. | DOI | MR | Zbl

[30] L. Rosier, Exact boundary controllability for the Korteweg-de Vries equation on a bounded domain. ESAIM: COCV 2 (1997) 33–55. | Numdam | MR | Zbl

[31] L. Rosier, Control of the surface of a fluid by a wavemaker. ESAIM: COCV 10 (2004) 346–380. | Numdam | MR | Zbl

[32] L. Rosier and B.-Y. Zhang, Global stabilization of the generalized Korteweg-de Vries equation. SIAM J. Control Optim. 45 (2006) 927–956. | DOI | MR | Zbl

[33] D.L. Russell and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation. Trans. Am. Math. Soc. 348 (1996) 3643–3672. | DOI | MR | Zbl

[34] J. Simon, Compact sets in the L^p(0; T; B) spaces. Anal. Mat. Pura Appl. 146 (1987) 65–96. | DOI | MR | Zbl

[35] E.C. Titchmarsh, The theory of functions. Oxford University Press, Oxford (1958). | JFM | MR | Zbl