On the Uniform Controllability of the Inviscid and Viscous Burgers-$\alpha$ Systems

TL;DR

This paper proves the global uniform exact controllability of both inviscid and viscous Burgers-$$ systems, using the return method and controllability results for regularized equations with filtered velocities.

Contribution

It establishes the first global uniform exact controllability results for Burgers-$$ systems, including viscous and inviscid cases, with novel control techniques.

Findings

Global uniform exact controllability for inviscid Burgers-$$ system.

Global exact controllability to constant states for viscous system.

Use of return method and controllability results for regularized equations.

Abstract

This work is devoted to prove the global uniform exact controllability of the inviscid and viscous Burgers-$\alpha$ systems. The state $y$ is the solution to a regularized Burgers equation, where the transport velocity $z$ consists of a filtered version of $y$ - specifically $z=(Id-\alpha^2\partial^2_ {xx})^{-1}y$ with $\alpha>0$ being a small parameter - in place of $y$. First, a global uniform exact controllability result for the nonviscous Burgers-$\alpha$ system with three scalar controls is obtained, using the return method. Then, global exact controllability to constant states of the viscous system is deduced from a local exact controllability result and a global approximate controllability result for smooth initial and target states.