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

Abstract

In this paper we study the global controllability of families of the so called non-viscous and viscous Burgers- 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 lter of characteristic wavelength . First, we prove a global exact controllability result (uniform with respect to ) for the non-viscous Burgers- system, using the return method and a xed-point argument. Then, the global uniform exact controllability to constant states is deduced for the viscous equations. To this purpose, we rst prove a local exact controllability property and, then, we establish a global approximate controllability result for smooth initial and target states.