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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
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

Primary 35Q53 Secondary 37K10 93B05 93D15 

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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

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