Geodesic
Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 78 (2023) no. 1, pp. 65-163

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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 elliptic functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Bibliography: 162 titles.
Keywords: optimal control, geometric control theory, left-invariant problems, sub-Riemannian geometry, Lie groups, optimal synthesis, elliptic functions. 
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     title = {Left-invariant optimal control problems on {Lie} groups that are integrable by elliptic functions},
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Yu. L. Sachkov. Left-invariant optimal control problems on Lie groups that are integrable by elliptic functions. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 78 (2023) no. 1, pp. 65-163. http://geodesic.mathdoc.fr/item/RM_2023_78_1_a1/