On the uniform controllability for a family of non-viscous and viscous Burgers-α systems

¹ Department of Mathematics, Federal University of Pernambuco, UFPE, CEP 50740-545, Recife, PE, Brazil
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: desouza@us.es

Received: 5 March 2021
Accepted: 24 June 2021

Abstract

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.