Control theory for the Burgers equation: Agrachev-Sarychev approach

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.