We consider evolution problems, such as diffusion convection equations or the linearized Navier-Stokes system, or a weak coupling of them, which we would like to "predict" on a time interval (T(0), T(0) + T) but for which, on one hand, the initial value of the state variable is unknown. On the other hand "measurements" of the solutions are known on a time interval (0, T(0)) and, for example, on a subdomain in the space variable. The classical approach in variational data assimilation is to look for the initial value at time 0, and this is known to be an ill-posed problem which has to be regularized. Here we propose to look for the value of the state variable at time T(0) ( the end time of the "measurements") and we prove on some basic examples that this is a well-posed problem. We give a result of exact reconstruction of the value at T(0) which is based on global Carleman inequalities, and we give an approximation algorithm which uses classical optimal control auxiliary problems. Using the same mathematical arguments, we also show why Tychonov regularization for variational data assimilation works in practical situations corresponding to realistic applications.