Exact controllability and stability of the Sixth Order Boussinesq equation

Abstract

The article studies the exact controllability and the stability of the sixth order Boussinesq equationon the interval S := [0, 2π] with periodic boundary conditions.It is shown that the system is locally exactly controllable in the classic Sobolev space, H s+3 (S) × H s (S) for s ≥ 0, for "small" initial and terminal states. It is also shown that if f is assigned as an internal linear feedback, the solution of the system is uniformly exponential decay to a constant state in H s+3 (S) × H s (S) for s ≥ 0 with "small" initial data assumption.

New Traveling Wave Solutions for the Sixth-order Boussinesq Equation

“…In addition to the aforementioned work, Christov, Maugin and Velarde [7] reexamined the Boussinesq-type equations for the shallow fluid layers and derived equation (1.1). The exact controllability and stability of the equation has been studied in [8]. However, the traveling wave solutions for (1.1) has not been considered.…”