Cost of controllability of the Burgers' equation linearized at a steady shock in the vanishing viscosity limit

Abstract: We consider the one-dimensional Burgers' equation linearized at a stationary shock, and investigate its null-controllability cost with a control at the left endpoint. We give an upper and a lower bound on the control time required for this cost to remain bounded in the vanishing viscosity limit, as well as a rough description of an admissible control. The proof relies on complex analysis and adapts methods previously used to tackle the same issue with a constant transport term.