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.
Highlights
In the case of the Burgers-α system, the local uniform null controllability of the viscous system (1.2) with distributed and boundary controls was studied in [1]; later, the results have been extended to any equation of the b-family in [27]
The goal is to prove the following approximate controllability result starting from sufficiently smooth initial data: Proposition 4.3
The goal of this section is to prove the local exact controllability to space-independent trajectories for the Burgers-α system, with controls and associated states uniformly bounded with respect to α in appropriate spaces
Summary
We are going to deal with situations that lead to new difficulties compared to previous works on nonlinear parabolic equations. In the case of the Burgers-α system, the local uniform null controllability of the viscous system (1.2) with distributed and boundary controls was studied in [1]; later, the results have been extended to any equation of the b-family in [27].
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