Fr. 07.07.2023, Oberseminar Geomathematik
Thema: Geometric mechanics meets optimal control. Referent: Prof. Dr. Sina Ober-Blöbaum ((Universität Paderborn) Raum: ENC-B 205, Zeit: 12:30 Uhr
Geometric mechanics meets optimal control.
Freitag, den 7. Juli 2023,
12:30 Uhr, Raum ENC-B 205 ,
Referent: Prof. Dr. Sina Ober-Blöbaum
Abstrakt: In this talk we will show how concepts from geometric mechanics can be used in analysing and numerically solving optimal control problems for mechanical systems. In the first part of the talk we introduce geometric integration methods. These are structure-preserving methods with the aim of reproducing the behaviour of a dynamical system as realistically as possible. In particular, when using structure-preserving methods for simulating dynamical systems, certain geometric properties of the system are inherited from the numerical solution. A special class of geometric integrators are so-called variational integrators, which preserve the symplectic form and momentum maps induced by symmetries and also have an excellent long-term energy behaviour. We will demonstrate how variational integrators can be used to numerically solve optimal control problems and thereby not only preserving the structure of the dynamical systems but also the symplectic structure of the optimal control problem. In the second part of the talk we consider optimal control problems with symmetries and discuss how symmetries can be exploited to make solution methods more efficient. In particular, recent results on relationships between symmetries in optimal control problems and turnpike properties of the optimal control problem are presented. The turnpike property characterizes a quasi-static behaviour of solutions of optimal control problems defined on a large time horizon. In this case, the solutions converge to a neighbourhood of a steady state and stay there for a major part of the time interval. In many practical examples in mechanical and biological systems, the convergence is not towards a steady state but some attracting trajectory instead. We show that for the mechanical systems admitting a symmetry with respect to a Lie group action, these trajectories correspond to the relative equilibria of the system.