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

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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