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

In this paper, we study the boundary controllability of the Gear–Grimshaw system posed on a finite domain (0, L), with Neumann boundary conditions: ⎧ ⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎩ ut + uux + uxxx + avxxx + a1vvx + a2(uv)x = 0, in (0, L)× (0, T ), cvt + rvx + vvx + abuxxx + vxxx + a2buux + a1b(uv)x = 0, in (0, L)× (0, T ), uxx(0, t) = h0(t), ux(L, t) = h1(t), uxx(L, t) = h2(t), in (0, T ), vxx(0, t) = g0(t), vx(L, t) = g1(t), vxx(L, t) = g2(t), in (0, T ), u(x, 0) = u0(x), v(x, 0) = v0(x), in (0, L). 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− 3 (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− 3 (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. Mathematics Subject Classification. Primary 35Q53; Secondary 37K10, 93B05, 93D15.