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Conjugate and cut loci of a two-sphere of revolution with application to optimal control. (English) Zbl 1184.53036
A result of Sinclair and Tanaka states: Given a smooth metric $$dr^2+ m^2(r^2)d\theta^2$$ on $$S^2$$, where $$r$$ is the angular variable along the meridian, $$\theta$$ is the angle of revolution, the reflective symmetry condition $$m(2a- r)= m(r)$$ holds, and the Gaussian curvature is monotone along ameridian, then (away from the pole) the cut locus of a point is a simple branch on the antipodal parallel. The authors of the present paper derive the same conclusion with sharper assumptions. These also give global optimal results is orbital transfer and for Lindblad equations in quantum control.

MSC:
 53C20 Global Riemannian geometry, including pinching 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 49K15 Optimality conditions for problems involving ordinary differential equations 70Q05 Control of mechanical systems
Loki
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References:
 [1] A. Agrachev, U. Boscain, M. Sigalotti, A Gauss-Bonnet like formula on two-dimensional almost Riemannian manifolds, Preprint SISSA 55/2006/M, 2006 · Zbl 1198.49041 [2] Bao, D.; Robles, C.; Shen, Z., Zermelo navigation on Riemannian manifolds, J. differential geom., 66, 3, 377-435, (2004) · Zbl 1078.53073 [3] Berger, M., A panoramic view of Riemannian geometry, (2003), Springer-Verlag Berlin · Zbl 1038.53002 [4] Bonnard, B.; Caillau, J.-B., Optimality results in orbit transfer, C. R. acad. sci. Paris, ser. I, 345, 319-324, (2007) · Zbl 1122.49018 [5] Bonnard, B.; Caillau, J.-B.; Dujol, R., Energy minimization of single-input orbit transfer by averaging and continuation, Bull. sci. math., 130, 8, 707-719, (2006) · Zbl 1136.93046 [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 linéaire, 24, 3, 395-411, (2007) · Zbl 1127.49017 [7] B. Bonnard, D. Sugny, Optimal control with applications in space and quantum dynamics, Preprint Institut math. Bourgogne, 2007 · Zbl 1266.49002 [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] Bonnard, B.; Chyba, M., Singular trajectories and their role in control theory, (2003), Springer-Verlag Berlin · Zbl 1022.93003 [10] Gluck, H.; Singer, D., Scattering of geodesic fields I and II, Ann. of math., 108, 347-372, (1978), and 110 (1979) 205-225 · Zbl 0399.58011 [11] Gravesen, J.; Markvorsen, S.; Sinclair, R.; Tanaka, M., The cut locus of a torus of revolution, Asian J. math., 9, 103-120, (2005) · Zbl 1080.53005 [12] Hebda, J.J., Metric structure of cut loci in surfaces and Ambrose’s problem, J. differential geom., 40, 621-642, (1994) · Zbl 0823.53031 [13] Itoh, J.I.; Kiyohara, K., The cut loci and the conjugate loci on ellipsoids, Manuscripta math., 114, 247-264, (2004) · Zbl 1076.53042 [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 [15] Myers, S.B., Connections between differential geometry and topology: I. simply connected surfaces and II. closed surfaces, Duke math. J., 1, 376-391, (1935), and 2 (1936) 95-102 · Zbl 0012.27502 [16] Poincaré, H., Sur LES lignes géodésiques des surfaces convexes, Trans. amer. math. soc., 6, 237-274, (1905) · JFM 36.0669.01 [17] Sakai, T., Riemannian geometry, Transl. math. monogr., vol. 149, (1996), American Mathematical Society Providence, RI [18] Shiohama, K.; Tanaka, M., Cut loci and distance spheres on Alexandrov surfaces, (), 531-560 · Zbl 0874.53032 [19] Shiohama, K.; Shioya, T.; Tanaka, M., The geometry of total curvature on complete open surfaces, Cambridge tracts in mathematics, vol. 159, (2003) · Zbl 1086.53056 [20] Sinclair, R.; Tanaka, M., A bound on the number of endpoints of the cut locus, LMS J. comput. math., 9, 21-39, (2006) · Zbl 1111.53004 [21] Sinclair, R.; Tanaka, M., The cut locus of a two-sphere of revolution and Toponogov’s comparison theorem, Tohoku math. J., 59, 2, 379-399, (2007) · Zbl 1158.53033 [22] Sugny, D.; Kontz, C.; Jauslin, H., Time-optimal control of a two-level dissipative quantum system, Phys. rev. A, 76, 023419, (2007)
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