Reference : Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem
Scientific journals : Article
Physical, chemical, mathematical & earth Sciences : Mathematics
Submersions, Hamiltonian systems and optimal solutions to the rolling manifolds problem
Grong, Erlend mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
SIAM Journal on Control & Optimization
Society for Industrial & Applied Mathematics
Yes (verified by ORBilu)
[en] Submersions ; Hamiltonian systems ; Rolling manifolds
[en] Given a submersion $\pi:Q \to M$ with an Ehresmann connection~$\calH$, we describe how to solve Hamiltonian systems on $M$ by lifting our problem to $Q$. Furthermore, we show that all solutions of these lifted Hamiltonian systems can be described using the original Hamiltonian vector field on $M$ along with a generalization of the magnetic force. This generalized force is described using the curvature of $\calH$ along with a new form of parallel transport of covectors vanishing on $\calH$. Using the Pontryagin Maximum Principle, we apply this theory to optimal control problems $M$ and $Q$ to get results on normal and abnormal extremals. We give a demonstration of our theory by considering the optimal control problem of one Riemannian manifold rolling on another without twisting or slipping along curves of minimal length.
Fonds National de la Recherche - FnR
FnR ; FNR7628746 > Anton Thalmaier > GEOMREV > Geometry of random evolutions > 01/03/2015 > 28/02/2018 > 2014

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