I³MS - Ober-Blöbaum Seminar
Prof. Dr. Sina Ober-Blöbaum - Variational Integration and Optimal Control: Theory and Applications
Computational Dynamics and Optimal Control, University of Paderborn
This talk focuses on different aspects of variational integrators and their use for numerical optimal control methods.
The key feature of variational integrators is that the time-stepping schemes are derived from a discrete variational principle based on a discrete action function that approximates the continuous one. The resulting schemes are structure preserving, i.e. they are symplectic-momentum conserving and exhibit good energy behaviour, meaning that no artificial dissipation is present and the energy error stays bounded over longterm simulations.
For the numerical solution of optimal control problems, direct methods are based on a discretization of the underlying differential equations which serve as equality constraints for the resulting finite dimensional nonlinear optimization problem. For the case of mechanical systems, the presented method, denoted by DMOC (Discrete Mechanics and Optimal Control), is based on the discretization of the variational structure of the system directly and thus inherits the structure preservation properties of variational integrators.
Different approaches are discussed to obtain integration schemes of higher accuracy such as higher order variational methods, time-adaptive schemes and multirate integrators. The main challenge for the construction of these schemes is to still guarantee the structure preserving properties which are typically destroyed using standard adaptive approaches. The applicability of the algorithms is demonstrated by different examples from mechanical and electrical engineering.