On the stabilization and controllability for a third order linear equation

We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation: $$ iu_t+i\gamma u_x+ \alpha u_{xx} + i\beta u_{xxx} =0, $$ where $u=u(x,t)$ is a complex valued function defined in $(0,L)\times(0,+\infty)$ and $\alpha$, $\beta$ and $\gamma$ are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths $L$, named $\mathcal{X}$, in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters $\alpha$, $\beta$ and $\gamma$.