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

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

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

Content Provider	Springer Nature Link
Author	Pazoto, Ademir F. Gallego, Fernando A. Capistra Filho, Roberto A.
Copyright Year	2016
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.
Ending Page	36
Page Count	36
Starting Page	1
File Format	PDF
ISSN	00442275
e-ISSN	14209039
Journal	Zeitschrift für angewandte Mathematik und Physik
Issue Number	5
Volume Number	67
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-08-13
Publisher Place	Cham
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.) Mathematical Methods in Physics KdV-like equations (Korteweg-de Vries) Gear–Grimshaw system Exact boundary controllability Critical set Dirichlet boundary conditions Neumann boundary conditions Controllability Stabilization of systems by feedback Theoretical and Applied Mechanics
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Physics and Astronomy

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