Optimal Control and Additive Perturbations Help in Estimating Ill-Posed and Uncertain Dynamical Systems

Ordinary differential equations (ODE) are routinely calibrated on real data for estimating unknown parameters or for reverse-engineering. Nevertheless, standard statistical techniques can give disappointing results because of the complex relationship between parameters and states, which makes the corresponding estimation problem ill-posed. Moreover, ODE are mechanistic models that are prone to modeling errors, whose influences on inference are often neglected during statistical analysis. We propose a regularized estimation framework, called Tracking, which consists in adding a perturbation (<i>L</i>² function) to the original ODE. This perturbation facilitates data fitting and represents also possible model misspecifications, so that parameter estimation is done by solving a trade-off between data fidelity and model fidelity. We show that the underlying optimization problem is an optimal control problem that can be solved by the Pontryagin maximum principle for general nonlinear and partially observed ODE. The same methodology can be used for the joint estimation of finite and time-varying parameters. We show, in the case of a well-specified parametric model that our estimator is consistent and reaches the root-<i>n</i> rate. In addition, numerical experiments considering various sources of model misspecifications shows that Tracking still furnishes accurate estimates. Finally, we consider semiparametric estimation on both simulated data and on a real data example. Supplementary materials for this article are available online.