Estimating Parameters in Optimal Control Problems

In nature, there are many processes that are assumed to run optimally, for example, human motion or, in particular, human gait. In this paper, we consider models of such processes, which can be described by optimal control problems. These models are characterized by quantities that are not known and cannot be measured (examples are system parameters or the constitution of the optimality criterion). Our goal is to determine these unknown quantities from measurement data, which is equivalent to solving a parameter estimation problem based on an optimal control model. We investigate two approaches for finding the solution of this hierarchical optimization problem; both follow the idea of replacing the underlying optimal control problem by its optimality conditions. The first approach is based on Pontryagin's maximum principle, whereas the second approach uses direct multiple shooting and the corresponding Karush--Kuhn--Tucker conditions. Both approaches pose numerical challenges that are discussed in this pa...