Geodesic
Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 1, pp. 99-163

Voir la notice de l'article provenant de la source Math-Net.Ru

Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed. Bibliography: 91 titles.
Keywords: optimal control, geometric control theory, left-invariant problems, sub-Riemannian geometry, Lie groups, optimal synthesis. 
@article{RM_2022_77_1_a3,
     author = {Yu. L. Sachkov},
     title = {Left-invariant optimal control problems on {Lie} groups: classification and problems integrable by elementary functions},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {99--163},
     publisher = {mathdoc},
     volume = {77},
     number = {1},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RM_2022_77_1_a3/}
}
TY  - JOUR
AU  - Yu. L. Sachkov
TI  - Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2022
SP  - 99
EP  - 163
VL  - 77
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RM_2022_77_1_a3/
LA  - en
ID  - RM_2022_77_1_a3
ER  - 
%0 Journal Article
%A Yu. L. Sachkov
%T Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2022
%P 99-163
%V 77
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RM_2022_77_1_a3/
%G en
%F RM_2022_77_1_a3
Yu. L. Sachkov. Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 77 (2022) no. 1, pp. 99-163. http://geodesic.mathdoc.fr/item/RM_2022_77_1_a3/