空间域上具有临界尺寸限制的Gear-Grimshaw系统的Neumann边界能控性
Neumann Boundary Controllability of the Gear--Grimshaw System With Critical Size Restrictions on the Spacial Domain
摘要：
本文研究了有限区域$(0,L)$上的Gear-Grimshaw系统的边界能控性，其Neumann边界条件为：\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}我们首先证明了当$h_2(t)=g_2(t)=0$满足时，相应的围绕原点的线性化系统在$(L^2(0,L))^2$中是精确能控的.在这种情况下，精确的可控性属性导出任何$L>0$与控制功能$h_0, g_0\in H^{-\frac{1}{3}}(0,T)$和$h_1, g_1\in L^2(0,T)$。如果我们改变控制的位置，并考虑$h_0(t)=h_2(t)=0$（分别$g_0(t)=g_2(t)=0)$我们用控制函数$g_0, g_2\in H^{-\frac{1}{3}}(0,T)$和$h_1, g_1\in L^2(0,T)$得到结果当且仅当空间域$(0,L)$的长度$L$属于可数集。在所有情况下，控制的规律性在时间上是尖锐的。如果只有一个控制在边界条件下起作用，则$h_0(t)=g_0(t)=h_2(t)=g_2(t)=0$和$g_1(t)=0$（分别为$h_1(t)=0$），线性化系统被证明是精确可控的小值的长度$L$和大时间的控制$T$。最后，通过压缩映射原理证明了非线性系统是局部精确能控的，如果相关的线性化系统是精确能控的。
英文摘要：
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.
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