Small-time global null controllability of generalized Burgers’ equations

Laboratoire Jacques-Louis Lions, Sorbonne Université, Paris, France

^* Corresponding author: remi.robin@inria.fr

Received: 8 June 2022
Accepted: 26 March 2023

Abstract

In this paper, we study the small-time global null controllability of the generalized Burgers’ equations y_t + γ|y|^γ-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 γ > 3/2 we prove that the system is small-time globally null controllable. Our proof relies on the return method and a careful analysis of the shape and dissipation of a boundary layer.