In this paper, we consider the back and forth nudging algorithm that has been introduced
for data assimilation purposes. It consists of iteratively and alternately solving forward
and backward in time the model equation, with a feedback term to the observations. We
consider the case of 1-dimensional transport equations, either viscous or inviscid, linear
or not (Burgers’ equation). Our aim is to prove some theoretical results on the
convergence, and convergence properties, of this algorithm. We show that for non viscous
equations (both linear transport and Burgers), the convergence of the algorithm holds
under observability conditions. Convergence can also be proven for viscous linear
transport equations under some strong hypothesis, but not for viscous Burgers’ equation.
Moreover, the convergence rate is always exponential in time. We also notice that the
forward and backward system of equations is well posed when no nudging term is considered.