On the local null controllability of a viscous Burgers' system in finite time

TL;DR

This paper investigates the local null controllability of a viscous Burgers' system, establishing that it cannot be controlled to zero in finite time, extending previous results about small-time controllability limitations.

Contribution

The paper proves that the viscous Burgers' system is not locally null controllable in finite time, providing a new proof inspired by methods used for the KdV system.

Findings

The system cannot be driven to zero in finite time.

The result extends known small-time controllability limitations.

The approach differs from previous proofs by Marbach.

Abstract

This paper is devoted to the local null controllability of the Burgers control system $y_t - y_{xx} + y y_x = u(t)$ on a bounded interval imposed by the zero Dirichlet boundary condition. It is known from the work of Marbach that this control system is not locally null controllable in small time. In this paper, we prove that the system is not locally null controllable in finite time as well. Our approach is inspired by the works of Coron, Koenig, and Nguyen, and Nguyen on the controllability of the KdV system and is different from the one of Marbach.