Let M be a differentiable manifold, T$$M be its tangent space at x ∈ M and TM = {(x, y);x ∈ M;y ∈ T$$M} be its tangent bundle. A C0-Finsler structure is a continuous function F : TM → [0, ∞) such that F(x, ⋅) : T$$M → [0, ∞) is an asymmetric norm. In this work we introduce the Pontryagin type C0-Finsler structures, which are structures that satisfy the minimum requirements of Pontryagin’s maximum principle for the problem of minimizing paths. We define the extended geodesic field ℰ on the slit cotangent bundle T*M\0 of (M, F), which is a generalization of the geodesic spray of Finsler geometry. We study the case where ℰ is a locally Lipschitz vector field. We show some examples where the geodesics are more naturally represented by ℰ than by a similar structure on TM. Finally we show that the maximum of independent Finsler structures is a Pontryagin type C0-Finsler structure where ℰ is a locally Lipschitz vector field.
Keywords: Geodesic field, extended geodesic field, cotangent bundle, Pontryagin’s maximum principle, Finsler structure, $$0-Finsler structure
@article{COCV_2022__28_1_A19_0,
author = {Rodrigues, Hugo Murilo and Fukuoka, Ryuichi},
title = {Geodesic fields for {Pontryagin} type $C^0${-Finsler} manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2022013},
mrnumber = {4387182},
zbl = {1485.49027},
language = {en},
url = {https://numdam.org/articles/10.1051/cocv/2022013/}
}
TY - JOUR AU - Rodrigues, Hugo Murilo AU - Fukuoka, Ryuichi TI - Geodesic fields for Pontryagin type $C^0$-Finsler manifolds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://numdam.org/articles/10.1051/cocv/2022013/ DO - 10.1051/cocv/2022013 LA - en ID - COCV_2022__28_1_A19_0 ER -
%0 Journal Article %A Rodrigues, Hugo Murilo %A Fukuoka, Ryuichi %T Geodesic fields for Pontryagin type $C^0$-Finsler manifolds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://numdam.org/articles/10.1051/cocv/2022013/ %R 10.1051/cocv/2022013 %G en %F COCV_2022__28_1_A19_0
Rodrigues, Hugo Murilo; Fukuoka, Ryuichi. Geodesic fields for Pontryagin type $C^0$-Finsler manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 19. doi: 10.1051/cocv/2022013
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