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

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: (Formulae presented.). We first prove that the corresponding linearized system around the origin is exactly controllable in (L2(0,L))2 when h2(t) = g2(t) = 0. In this case, the exact controllability property is derived for any L > 0 with control functions h0,g0∈H-13(0,T) and h1, g1∈ L2(0 , T). If we change the position of the controls and consider h0(t) = h2(t) = 0 (resp. g0(t) = g2(t) = 0) , we obtain the result with control functions g0,g2∈H-13(0,T) and h1, g1∈ L2(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, h0(t) = g0(t) = h2(t) = g2(t) = 0 and g1(t) = 0 (resp. h1(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.