A regularization method for the parameter estimation problem in ordinary differential equations via discrete optimal control theory

We present a parameter estimation method in Ordinary Differential Equation (ODE) models.Due to complex relationships between parameters and statesthe use of standard techniquessuch as nonlinear least squares can lead to the presence of poorly identifiable parameters.Moreover, ODEs are generally approximations of the true process and the influence of mis-specification on inference is often neglected. Methods based on control theory have emergedto regularize the ill posed problem of parameter estimationin this context. However, they arecomputationally intensive and rely on a nonparametric state estimator known to be biased inthe sparse sample case. In this paper, we construct criteriabased on discrete control theorywhich are computationally efficient and bypass the presmoothing step of signal estimationwhile retaining the benefits of control theory for estimation. We describe how the estimationproblem can be turned into a control one and present the numerical methods used to solve it.We show convergence of our estimator in the parametric and well-specified case. For smallsample sizes, numerical experiments with models containing poorly identifiable parametersand with various sources of model misspecification demonstrate the acurracy of our method.We finally test our approach on a real data example