On the control of the Burgers-alpha model

This work is devoted to prove the local null controllability of the Burgers-$\alpha$ model. The state is the solution to a regularized Burgers equation, where the transport term is of the form $z{y}_x$, $z=(Id-{\alpha }^2\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}{)}^{-1}y$ and $\alpha >0$ is a small parameter. We also prove some results concerning the behavior of the null controls and associated states as $\alpha \to 0^{+}$.