  The paper is devoted to studying the 1D viscous Burgers equation controlled
by an external force. It is assumed that the initial state is essentially
bounded, with no decay condition at infinity, and the control is a
trigonometric polynomial of low degree with respect to the space variable. We
construct explicitly a control space of dimension 11 that enables one to steer
the system to any neighbourhood of a given final state in local topologies. The
proof of this result is based on an adaptation of the Agrachev-Sarychev
approach to the case of an unbounded domain.
