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arXiv · 2006.05335

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

Raul K. C. Araújo·Enrique Fernández-Cara·Diego A. Souza
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Abstract

This work is devoted to prove the global uniform exact controllability of the inviscid and viscous Burgers-$α$ 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 $z=(Id-α^2\partial^2_ {xx})^{-1}y$ with $α>0$ being a small parameter - in place of $y$. First, a global uniform exact controllability result for the nonviscous Burgers-$α$ 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.

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Raul K. C. Araújo, Enrique Fernández-Cara, Diego A. Souza. 2020-06-09. On the Uniform Controllability of the Inviscid and Viscous Burgers-$α$ Systems. https://doi.org/10.1051/cocv%2F2021073

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