Neumann boundary controllability of the Gear–Grimshaw system with critical size restrictions on the spatial domain

  • Roberto A. Capistrano-FilhoEmail author
  • Fernando A. Gallego
  • Ademir F. Pazoto


In this paper, we study the boundary controllability of the Gear–Grimshaw system posed on a finite domain (0, L), with Neumann boundary conditions:
$$\label{abs}\begin{cases}u_{t} + uu_x+u_{xxx} + a v_{xxx} + a_1vv_x+a_2 (uv)_x =0, & \text{in}\,\, (0,L)\times (0,T),\\ c v_t +rv_x +vv_x+abu_{xxx} +v_{xxx}+a_2buu_x+a_1b(uv)_x =0, &\text{in} \,\, (0,L)\times (0,T),\\ u_{xx}(0,t)=h_0(t),\,\,u_x(L,t)=h_1(t),\,\,u_{xx}(L,t)=h_2(t), &\text{in} \,\, (0,T),\\ v_{xx}(0,t)=g_0(t),\,\,v_x(L,t)=g_1(t),\,\,v_{xx}(L,t)=g_2(t), &\text{in} \,\, (0,T),\\ u(x,0)= u^0(x), \quad v(x,0)= v^0(x), & \text{in} \,\, (0,L).\nonumber\end{cases}$$
We first prove that the corresponding linearized system around the origin is exactly controllable in \({\left(L^2(0,L)\right)^2}\) when h 2(t) = g 2(t) = 0. In this case, the exact controllability property is derived for any L >  0 with control functions \({h_0, g_0 \in H^{-\frac{1}{3}}(0,T)}\) and \({h_1, g_1\in L^2(0,T)}\). If we change the position of the controls and consider \({h_0(t)=h_2(t)=0}\) (resp. \({g_0(t)=g_2(t)=0)}\), we obtain the result with control functions \({g_0, g_2\in H^{-\frac{1}{3}}(0,T)}\) and \({h_1, g_1\in L^2(0,T)}\) if and only if the length L of the spatial domain (0, L) does not belong to a countable set. In all cases, the regularity of the controls are sharp in time. If only one control act in the boundary condition, \({h_0(t)=g_0(t)=h_2(t)=g_2(t)=0}\) and g 1(t) = 0 (resp. h 1(t) = 0), the linearized system is proved to be exactly controllable for small values of the length L and large time of control T. Finally, the nonlinear system is shown to be locally exactly controllable via the contraction mapping principle, if the associated linearized systems are exactly controllable.


Gear–Grimshaw system Exact boundary controllability Neumann boundary conditions Dirichlet boundary conditions Critical set 

Mathematics Subject Classification

Primary 35Q53 Secondary 37K10 93B05 93D15 


  1. 1.
    Bona J.L., Ponce G., Saut J.-C., Tom M.M.: A model system for strong interaction between internal solitary waves. Commun. Math. Phys. 143, 287–313 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bona J.J., Sun S.M., Zhang B.-Y.: A nonhomogeneous boundary-value problem for the Korteweg–de Vries equation posed on a finite domain. Commun. Partial Differ. Equ. 28, 1391–1438 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Caicedo, M., Capistrano-Filho, R.A., Zhang, B.: Neumann boundary controllability of the Korteweg–de Vries equation on a bounded domain. Preprint (2015). arXiv:1508.07525
  4. 4.
    Capistrano-Filho, R.A., Sun, S.M., Zhang, B.-Y.: General boundary value problems of the Korteweg–de Vries equation on a bounded domain. Preprint (2016)Google Scholar
  5. 5.
    Cerpa E., Pazoto A.F.: A note on the paper On the controllability of a coupled system of two Korteweg–de Vries equations. Commun. Contemp. Math. 13, 183–189 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cerpa E., Rivas I., Zhang B.-Y.: Boundary controllability of the Korteweg–de Vries equation on a bounded domain. SIAM J. Control Optim. 51, 2976–3010 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dolecki S., Russell D.L.: A general theory of observation and control. SIAM J. Control Optim. 15, 185–220 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gear J.A., Grimshaw R.: Weak and strong interactions between internal solitary waves. Stud. Appl. Math. 70, 235–258 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lions, J.-L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Masson, Paris (1988)Google Scholar
  10. 10.
    Rosier L.: Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain. ESAIM Control Optim. Cal. Var. 2, 33–55 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Micu, S., Ortega, J.H.: On the controllability of a linear coupled system of Korteweg–de Vries equations. In: Mathematical and Numerical Aspects of Wave Propagation (Santiago de Compostela, 2000), pp. 1020–1024. SIAM, Philadelphia (2000)Google Scholar
  12. 12.
    Micu S., Ortega J., Pazoto A.: On the controllability of a coupled system of two Korteweg–de Vries equation. Commun. Contemp. Math. 11(5), 779–827 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kenig C.E., Ponce G., Vega L.: Well-posedness of the initial value problem for the Korteweg–de Vries equation. J. Am. Math. Soc. 4, 323–347 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Saut J.-C., Tzvetkov N.: On a model system for the oblique interaction of internal gravity waves. M2AN Math. Model. Numer. Anal. 34, 501–523 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yosida K.: Functional Analysis. Springer, Berlin (1978)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Roberto A. Capistrano-Filho
    • 1
    Email author
  • Fernando A. Gallego
    • 2
  • Ademir F. Pazoto
    • 2
  1. 1.Department of MathematicsFederal University of PernambucoRecifeBrazil
  2. 2.Institute of MathematicsFederal University of Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations