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Ordinary Differential Equations are widespread tools to model chemical,physical, biological process but they usually rely on parameters which are ofcritical importance in terms of dynamic and need to be estimated directly fromthe data. Classical statistical approaches (nonlinear least squares, maximumlikelihood estimator) can give unsatisfactory results because of computationaldifficulties and ill-posedness of the statistical problem. New estimationmethods that use some nonparametric devices have been proposed to circumventthese issues. We present a new estimator that shares properties with Two-Stepestimator and Generalized Smoothing (introduced by Ramsay et al, 2007). Weintroduce a perturbed model and we use optimal control theory for constructinga criterion that aims at minimizing the discrepancy with data and the model.Here, we focus on the case of linear Ordinary Differential Equations as ourcriterion has a closed-form expression that permits a detailed analysis. Ourapproach avoids the use of a nonparametric estimator of the derivative, whichis one of the main cause of inaccuracy in Two-Step estimators. Moreover, wetake into account model discrepancy and our estimator is more robust to modelmisspecification than classical methods. The discrepancy with the parametricODE model correspond to the minimum perturbation (or control) to apply to theinitial model. Its qualitative analysis can be informative for misspecificationdiagnosis. In the case of well-specified model, we show the consistency of ourestimator and that we reach the parametric root-n rate when regression splinesare used in the first step.

A Tracking Approach to Parameter Estimation in Linear Ordinary Differential Equations