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An optimal route on a sub-Riemannian manifold does not make sharp turns

New restrictions on the shape of optimal paths

University of Jyväskylä - Jyväskylän yliopisto

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IMAGE: In his thesis, FM Eero Hakavuori gives new restrictions on the shape of optimal paths. view more 

Credit: University of Jyvaskyla

The regularity of optimal routes on sub-Riemannian manifolds has been an important open problem in sub-Riemannian geometry since the early 90s. In his thesis, FM Eero Hakavuori gives new restrictions on the shape of optimal paths. The most important new restriction is the lack of sharp turns, i.e., corners.

Sub-Riemannian geometry has application in robotics in the control of mechanical systems. For such control systems, the lack of sharp turns is helpful.

"Usually a practical mechanism cannot suddenly change its directions, but instead requires some transition period", remarks Hakavuori.

In his thesis Hakavuori studies in particular so called Carnot groups. Carnot groups contain information about the small scall structure of sub-Riemannian manifolds and help answer to questions about the local behavior of optimal paths.

In addition to questions about local behavior, the thesis also studies the large scale behavior of optimal paths. One of the obtained results is that an optimal path cannot roam freely on a Carnot group, but must instead be concentrated near a lower dimensional subspace.

The thesis has been published in the University of Jyväskylä's JYU Dissertations -series number 103, Jyväskylä 2019, ISSN 2489-9003, ISBN: 978-951-39-7810-5.

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Link to publication in the JYX-archive: http://urn.fi/URN:ISBN:978-951-39-7810-5 (pdf)

M.Sc. Eero Hakavuori defends his doctoral dissertation in Subject "Sub-Riemannian geodesics" on Saturday 10th August 2019 at 12:00 in the lecture hall MaA211 at Mattilanniemi. Opponent Professor is Mario Sigalotti (Inria Paris and LJLL, Université Sorbonne, France) and Custos is Dr. Enrico Le Donne, Academy Research Fellow from University of Jyväskylä. The doctoral dissertation is held in English.

For further information:

M.Sc. Eero Hakavuori, eero.j.hakavuori@jyu.fi

Communications officer, Tanja Heikkinen, tanja.s.heikkinen@jyu.fi, tel. +358 50 581 8351

The Faculty on Mathematics and Science

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