Small-time global null controllability of generalized Burgers' equations

TL;DR

This paper proves that the generalized Burgers' equations are globally null controllable in small time using boundary controls, under certain conditions on the nonlinearity parameter, employing the return method and boundary layer analysis.

Contribution

It establishes small-time global null controllability for generalized Burgers' equations with b3 > 3/2, a result not previously known for this class of equations.

Findings

System is small-time globally null controllable for b3 > 3/2

Return method effectively used for controllability proof

Boundary layer analysis critical for dissipation understanding

Abstract

In this paper, we study the small-time global null controllability of the generalized Burgers' equations $y_t + \gamma |y|^{\gamma-1}y_x-y_{xx}=u(t)$ 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 $\gamma>3/2$ 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.