Neumann Boundary Controllability of the Gear--Grimshaw System With Critical Size Restrictions on the Spacial Domain

TL;DR

This paper investigates the boundary controllability of the Gear-Grimshaw system with Neumann boundary conditions, establishing exact controllability results for linearized and nonlinear cases under various boundary control configurations and domain sizes.

Contribution

It provides new controllability results for the Gear-Grimshaw system, including conditions on domain length and control placement, extending understanding of boundary control for coupled dispersive systems.

Findings

Linearized system is exactly controllable when certain boundary controls are active.

Controllability depends on the domain length being in a countable set for some control configurations.

Nonlinear system is locally exactly controllable under the same conditions as the linearized system.

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: \begin{equation} \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} \end{equation} We first prove that the corresponding linearized system around the origin is exactly controllable in $(L^2(0,L))^2$ 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 $(0,L)$ belongs 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.