  This paper is devoted to a description of a general approach introduced by
Agrachev and Sarychev in 2005 for studying some control problems for
Navier-Stokes equations. The example of a 1D Burgers equation is used to
illustrate the main ideas. We begin with a short discussion of the Cauchy
problem and establish a continuity property for the resolving operator. We next
turn to the property of approximate controllability and prove that it can be
achieved by a two-dimensional external force. Finally, we investigate a
stronger property, when the approximate controllability and the exact
controllability of finite-dimensional functionals are proved simultaneously.
