Small-time global null controllability of generalized Burgers' equations

In this paper, we study the small-time global null controllability of the generalized Burgers' equations $y_t + \gamma |y|^{\gamma-1}y_x-y_{xx}=u(t)$ on the segment $[0,1]$. The scalar control $u(t)$ is uniform in space and plays a role similar to the pressure in higher dimension. We set a right Dirichlet boundary condition $y(t,1)=0$, and allow a left boundary control $y(t,0)=v(t)$. Under the assumption $\gamma>3/2$ we prove that the system is small-time global null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer.