Small-time global null controllability of generalized Burgers' equations

Rémi Robin (1, 2)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 CaGE - Control And GEometry

In this paper, we study the small-time global null controllability of the generalized Burgers' equations $y_t + (|y|^\gamma)_x-y_{xx}=u(t)$ 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.

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Rémi Robin. Small-time global null controllability of generalized Burgers' equations. 2022. ⟨hal-03690584v1⟩