Conjugate and cut loci of a two-sphere of revolution with application to optimal control

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1. [1] A. Agrachev, U. Boscain, M. Sigalotti, A Gauss–Bonnet like formula on two-dimensional almost Riemannian manifolds, Preprint SISSA 55/2006/M, 2006. Zbl1198.49041MR2384632
2. [2] Bao D., Robles C., Shen Z., Zermelo navigation on Riemannian manifolds, J. Differential Geom.66 (3) (2004) 377-435. Zbl1078.53073MR2106471
3. [3] Berger M., A Panoramic View of Riemannian Geometry, Springer-Verlag, Berlin, 2003. Zbl1038.53002MR2002701
4. [4] Bonnard B., Caillau J.-B., Optimality results in orbit transfer, C. R. Acad. Sci. Paris, Ser. I345 (2007) 319-324. Zbl1122.49018MR2359089
5. [5] Bonnard B., Caillau J.-B., Dujol R., Energy minimization of single-input orbit transfer by averaging and continuation, Bull. Sci. Math.130 (8) (2006) 707-719. Zbl1136.93046MR2276200
6. [6] Bonnard B., Caillau J.-B., Riemannian metric of the averaged energy minimization problem in orbital transfer with low thrust, Ann. Inst. H. Poincaré Anal. Non Linéaire24 (3) (2007) 395-411. Zbl1127.49017MR2319940
7. [7] B. Bonnard, D. Sugny, Optimal control with applications in space and quantum dynamics, Preprint Institut math. Bourgogne, 2007. Zbl1266.49002
8. [8] B. Bonnard, J.B. Caillau, M. Tanaka, One-parameter family of Clairaut–Liouville metrics with application to optimal control, HAL preprint No 00177686 (2007), url: hal.archives-ouvertes.fr/hal-00177686. 
9. [9] Bonnard B., Chyba M., Singular Trajectories and Their Role in Control Theory, Springer-Verlag, Berlin, 2003. Zbl1022.93003MR1996448
10. [10] Gluck H., Singer D., Scattering of geodesic fields I and II, Ann. of Math.108 (1978) 347-372, and 110 (1979) 205–225. Zbl0436.53047MR506991
11. [11] Gravesen J., Markvorsen S., Sinclair R., Tanaka M., The cut locus of a torus of revolution, Asian J. Math.9 (2005) 103-120. Zbl1080.53005MR2150694
12. [12] Hebda J.J., Metric structure of cut loci in surfaces and Ambrose's problem, J. Differential Geom.40 (1994) 621-642. Zbl0823.53031MR1305983
13. [13] Itoh J.I., Kiyohara K., The cut loci and the conjugate loci on ellipsoids, Manuscripta Math.114 (2004) 247-264. Zbl1076.53042MR2067796
14. [14] K. Fleischmann, Die geodätischen Linien auf Rotationsflächen, Liegnitz, Dissertation, Seyffarth, 1915. Persistent URL: http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN317487884. JFM45.1372.05
15. [15] Myers S.B., Connections between Differential Geometry and Topology: I. Simply connected surfaces and II. Closed surfaces, Duke Math. J.1 (1935) 376-391, and 2 (1936) 95–102. Zbl0012.27502MR1545884JFM61.0834.03
16. [16] Poincaré H., Sur les lignes géodésiques des surfaces convexes, Trans. Amer. Math. Soc.6 (1905) 237-274. Zbl36.0669.01MR1500710JFM36.0669.01
17. [17] Sakai T., Riemannian Geometry, Transl. Math. Monogr., vol. 149, American Mathematical Society, Providence, RI, 1996. Zbl0886.53002MR1390760
18. [18] Shiohama K., Tanaka M., Cut loci and distance spheres on Alexandrov surfaces, in: Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Collection Soc. Math. France, vol. 1, Séminaires & Congrès, 1996, pp. 531-560. Zbl0874.53032MR1427770
19. [19] Shiohama K., Shioya T., Tanaka M., The Geometry of Total Curvature on Complete Open Surfaces, Cambridge Tracts in Mathematics, vol. 159, 2003. Zbl1086.53056MR2028047
20. [20] Sinclair R., Tanaka M., A bound on the number of endpoints of the cut locus, LMS J. Comput. Math.9 (2006) 21-39. Zbl1111.53004MR2199583
21. [21] Sinclair R., Tanaka M., The cut locus of a two-sphere of revolution and Toponogov's comparison theorem, Tohoku Math. J.59 (2) (2007) 379-399. Zbl1158.53033MR2365347
22. [22] Sugny D., Kontz C., Jauslin H., Time-optimal control of a two-level dissipative quantum system, Phys. Rev. A76 (2007) 023419. 

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