Data Assimilation is important in meteorology and oceanography, because it is a way to improve the models with newly measured data, statically or dynamically. It is a type of inverse problem for which the most popular solution method is least square with regularization and optimal control algorithms. As control theory assumes differentiability, there are mathematical difficulties when viscosity is neglected and the modeling uses a conservation law like the shallow water or Euler equations. In this paper we study the differentiated equations of some systems of conservation laws and show that Calculus of Variation can be applied in a formal and rigorous manner provided that principal values are defined at shocks and equations written in the sense of distribution theory. Numerical illustrations are given for the control of shocks for Burgers’ equation and for the shallow water equations in one space dimension.
"Data Assimilation for Conservation Laws." Methods Appl. Anal. 12 (2) 103 - 134, June2005.