Dubins' problem on surfaces II: Nonpositive curvature

SIAM Journal on Control and Optimization

Volumn 45, Issue 2, 2006, Pages 457-482

  Sigalotti, Mario ^a   Chitour, Yacine ^b  

^a INRIA   (France)

^b UNIV PARIS SUD   (France)

Author keywords

Controllability; Geodesic curvature; Geometric control; Nonpositive curvature

Indexed keywords

CONTROLLABILITY; INTEGRAL EQUATIONS; PROBLEM SOLVING;

GEODESIC CURVATURE; GEOMETRIC CONTROL; NONPOSITIVE CURVATURE;

OPERATIONS RESEARCH;

EID: 33947242429     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/040619739     Document Type: Article

References (31)

1. A. A. AGRACHEV AND Y. L. SACHKOV Control Theory from the Geometric Viewpoint, Encyclopaedia Math. Sci., 87, Springer-Verlag, Berlin, 2004.
2. 3, SIAM J. Control Optim., 42 (2003), pp. 513-531.
3. W. BALLMANN, Axial isometries of manifolds of non-positive curvature, Math. Ann., 259 (1982), pp. 131-144.
4. W. BALLMANN, M. GROMOV, AND V. SCHROEDER, Manifolds of Nonpositive Curvature, Progr. Math., 61, Birkhäuser, Boston, 1985.
5. U. BOSCAIN AND Y. CHITOUR, Time optimal synthesis for left-invariant control systems on SO(3), SIAM J. Control Optim., to appear.
6. Y. CHITOUR, Applied and Theoretical Aspects of the Controllability of Nonholonomic Control System, Ph.D. Thesis, Rutgers University, New Brunswick, NJ, 1996.
7. Y. CHITOUR AND M. SIGALOTTI, Dubins' problem on surfaces. I. Nonnegative curvature, J. Geom. Anal., 15 (2005), pp. 565-587.
8. S. COHN-VOSSEN, Kürzeste Wege und Totalkrümmung auf Flächen, Compos. Math., 2 (1935), pp. 69-133.
9. L. E. DUBINS, On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents, Amer. J. Math., 79 (1957), pp. 497-516.
10. P. EBERLEIN AND B. O'NEILL, Visibility manifolds, Pacific J. Math., 46 (1973), pp. 45-110.
11. A. HUBER, On subharmonic functions and differential geometry in the large, Comment. Math. Helv., 32 (1957), pp. 13-72.
12. V. JURDJEVIC, Geometric Control Theory, Cambridge Stud. Adv. Math. 52, Cambridge University Press, Cambridge, UK, 1997.
13. W. KLINGENBERG, A Course in Differential Geometry, Graduate Texts in Math. 51, Springer-Verlag, New York, Heidelberg, 1978.
14. A. KRENER, A generalization of Chow's theorem and the bang-bang theorem to non-linear control problems, SIAM J. Control, 12 (1974), pp. 43-51.
15. C. LOBRY, Contrôlabilité des systèmes non linéaires, SIAM J. Control, 8 (1970), pp. 573-605.
16. L. A. LYUSTERNIK AND A. I. FET, Variational problems on closed manifolds, Dokl. Akad. Nauk SSSR (N.S.), 81 (1951), pp. 17-18 (in Russian).
17. D. MITTENHUBER, Dubins' problem in the hyperbolic plane using the open disc model, in Geometric Control and Non-Holonomic Mechanics (Mexico City, 1996), CMS Conf. Proc. 25, Amer. Math. Soc., Providence, RI, 1998, pp. 115-152.
18. D. MITTENHUBER Dubins' problem is intrinsically three-dimensional, ESAIM Control Optim. Calc. Var., 3 (1998), pp. 1-22.
19. F. MONROY-PEREZ, Three-dimensional non-Euclidean Dubins' problem, in Geometric Control and Non-Holonomic Mechanics (Mexico City, 1996), CMS Conf. Proc. 25, Amer. Math. Soc., Providence, RI, 1998, pp. 153-181.
20. R. N. MURRAY, Z. X. LI, AND S. S. SASTRY, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL, 1994.
21. J. C. RATCLIFFE, Foundations of Hyperbolic Manifolds, Graduate Texts in Math. 149, Springer-Verlag, New York, 1994.
22. J. A. REEDS AND L. A. SHEPP, Optimal paths for a car that goes both forwards and backwards, Pacific J. Math., 145 (1990), pp. 367-393.
23. T. SAKAI, Riemannian Geometry, Transl. Math. Monogr. 149, Amer. Math. Soc., Providence, RI, 1996.
24. 3, SIAM J. Control Optim., 26 (1988), pp. 186-204.
25. H. SCHÄTTLER The local structure of time-optimal trajectories in dimension three under generic conditions, SIAM J. Control Optim., 26 (1988), pp. 899-918.
26. U. SERRES, On the curvature of two-dimensional optimal control systems and Zermelo's navigation problem, J. Math. Sci. to appear.
27. T. SHIOYA, Behavior of distant maximal geodesies infinitely connected complete 2-dimensional Riemannian manifolds, Mem. Amer. Math. Soc., 108, Providence, RI, 1994.
28. M. SPIVAK, A Comprehensive Introduction to Differential Geometry, Brandeis University, Waltham, MA, 1970.
29. H. J. SUSSMANN, Shortest 3-dimensional paths with a prescribed curvature bound, in Proc. IEEE CDC (New Orleans, 1995), IEEE Publications, New York, 1995, pp. 3306-3312.
30. H. J. SUSSMANN AND G. TANG, Shortest Paths for the Reeds-Shepp Car: A Worked Out Example of the Use of Geometric Techniques in Nonlinear Optimal Control, Rutgers Center for Systems and Control Technical Report 91-10, Rutgers University, New Brunswick, 1991.
31. G. THORBERGSSON, Closed geodesies on non-compact Riemannian manifolds, Math. Z., 159 (1978), pp. 249-258.