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
Oberseminar Geomathematik
Geometric mechanics meets optimal control.
Freitag, den 7. Juli 2023,
12:30 Uhr, Raum ENC-B 205 ,
Referent: Prof. Dr. Sina Ober-Blöbaum
(Universität Paderborn)
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.