Monitoring dynamical processes requires the estimation of the entire state, which is only partly accessible by measurements. Most quantities must be determined via model based state estimation, which in general is an ill-posed inverse problem. Regularization techniques have to be applied. To avoid undesired bias we omit the commonly used approach of regularizing the unknown initial data. To the author's knowledge the resulting minimization problem has not been analysed mathematically yet, which is the purpose of this paper. The first order necessary conditions are presented and the optimization problem is reduced by several variables. Hence, e.g. one otherwise necessary regularization parameter is dispensable. The influence of the regularization parameters and of the model matrices is studied in detail for linear models. It is shown that for any choice of regularization parameters the condition numbers of the evolving operators can be arbitrarily large, if the spectral radius of the system matrix is large. In the case of one state only we derive additionally bounds for the perturbation of the initial data resulting from measurement errors.