  Ordinary Differential Equations are a simple but powerful framework for
modeling complex systems. Parameter estimation from times series can be done by
Nonlinear Least Squares (or other classical approaches), but this can give
unsatisfactory results because the inverse problem can be ill-posed, even when
the differential equation is linear.
  Following recent approaches that use approximate solutions of the ODE model,
we propose a new method that converts parameter estimation into an optimal
control problem: our objective is to determine a control and a parameter that
are as close as possible to the data. We derive then a criterion that makes a
balance between discrepancy with data and with the model, and we minimize it by
using optimization in functions spaces: our approach is related to the
so-called Deterministic Kalman Filtering, but different from the usual
statistical Kalman filtering. e show the root-$n$ consistency and asymptotic
normality of the estimators for the parameter and for the states. Experiments
in a toy model and in a real case shows that our approach is generally more
accurate and more reliable than Nonlinear Least Squares and Generalized
Smoothing, even in misspecified cases.
