Exact controllability and stability of the Sixth Order Boussinesq equation

The article studies the exact controllability and the stability of the sixth order Boussinesq equation $$u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f, \quad \beta=\pm1,$$ on the interval $S:=[0,2\pi]$ with periodic boundary conditions. It is shown that the system is locally exactly controllable in the classic Sobolev space, $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$, 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 $H^{s+3}(S)\times H^s(S)$ for $s\geq 0$ with "small" initial data assumption.