Volume 27, 2021
|Number of page(s)||26|
|Published online||23 July 2021|
On the uniform controllability for a family of non-viscous and viscous Burgers-α systems
Department of Mathematics, Federal University of Pernambuco, UFPE, CEP 50740-545,
2 University of Sevilla, Dpto. E.D.A.N, Aptdo 1160, 41080 Sevilla, Spain
* Partially supported by CNPq (Brazil).
** Partially supported by grant MTM2016-76990-P, Ministry of Economy and Competitiveness (Spain).
*** Corresponding author: firstname.lastname@example.org
Accepted: 24 June 2021
In this paper we study the global controllability of families of the so called non-viscous and viscous Burgers-α systems by using boundary and space independent distributed controls. In these equations, the usual convective velocity of the Burgers equation is replaced by a regularized velocity, induced by a Helmholtz filter of characteristic wavelength α. First, we prove a global exact controllability result (uniform with respect to α) for the non-viscous Burgers-α system, using the return method and a fixed-point argument. Then, the global uniform exact controllability to constant states is deduced for the viscous equations. To this purpose, we first prove a local exact controllability property and, then, we establish a global approximate controllability result for smooth initial and target states.
Mathematics Subject Classification: 93B05 / 35Q35 / 35G25
Key words: Burgers-α system / global exact controllability / return method
© The authors. Published by EDP Sciences, SMAI 2021
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