On the Uniform Controllability of the Inviscid and Viscous Burgers-$α$ Systems
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| Format: | Preprint |
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2020
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| _version_ | 1866911769043140608 |
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| author | Araújo, Raul K. C. Fernández-Cara, Enrique Souza, Diego A. |
| author_facet | Araújo, Raul K. C. Fernández-Cara, Enrique Souza, Diego A. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2006_05335 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Uniform Controllability of the Inviscid and Viscous Burgers-$α$ Systems Araújo, Raul K. C. Fernández-Cara, Enrique Souza, Diego A. Analysis of PDEs 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. |
| title | On the Uniform Controllability of the Inviscid and Viscous Burgers-$α$ Systems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2006.05335 |