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

TL;DR

This paper analyzes the controllability cost of the linearized Burgers' equation at a shock, providing bounds and explicit controls as viscosity vanishes, with extensions to boundary control scenarios.

Contribution

It offers new bounds and explicit control constructions for the controllability cost of the linearized Burgers' equation at a shock, extending previous methods to this context.

Findings

Bounds on control time in vanishing viscosity limit

Explicit admissible control with limit behavior

Extension to boundary control on both endpoints

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, and construct an admissible control with an explicit limit behavior. We also provide an extension of the analysis to the case where the control acts on both endpoints. The proof relies on complex analysis and adapts methods previously used to tackle the same issue with a constant transport term.