  This paper is focused on the Morozov's principle applied to an abstract data
assimilation framework, with particular attention to three simple examples: the
data assimilation problem for the Laplace equation, the Cauchy problem for the
Laplace equation and the data assimilation problem for the heat equation. Those
ill-posed problems are regularized with the help of a mixed type formulation
which is proved to be equivalent to a Tikhonov regularization applied to a
well-chosen operator. The main issue is that such operator may not have a dense
range, which makes it necessary to extend well-known results related to the
Morozov's choice of the regularization parameter to that unusual situation. The
solution which satisfies the Morozov's principle is computed with the help of
the duality in optimization, possibly by forcing the solution to satisfy given
a priori constraints. Some numerical results in two dimensions are proposed in
the case of the data assimilation problem for the Laplace equation.
