  Ordinary Differential Equations are widespread tools to model chemical,
physical, biological process but they usually rely on parameters which are of
critical importance in terms of dynamic and need to be estimated directly from
the data. Classical statistical approaches (nonlinear least squares, maximum
likelihood estimator) can give unsatisfactory results because of computational
difficulties and ill-posedness of the statistical problem. New estimation
methods that use some nonparametric devices have been proposed to circumvent
these issues. We present a new estimator that shares properties with Two-Step
estimator and Generalized Smoothing (introduced by Ramsay et al, 2007). We
introduce a perturbed model and we use optimal control theory for constructing
a criterion that aims at minimizing the discrepancy with data and the model.
Here, we focus on the case of linear Ordinary Differential Equations as our
criterion has a closed-form expression that permits a detailed analysis. Our
approach avoids the use of a nonparametric estimator of the derivative, which
is one of the main cause of inaccuracy in Two-Step estimators. Moreover, we
take into account model discrepancy and our estimator is more robust to model
misspecification than classical methods. The discrepancy with the parametric
ODE model correspond to the minimum perturbation (or control) to apply to the
initial model. Its qualitative analysis can be informative for misspecification
diagnosis. In the case of well-specified model, we show the consistency of our
estimator and that we reach the parametric root-n rate when regression splines
are used in the first step.
