Dubins' problem is intrinsically three-dimensional

Dirk Mittenhuber

ESAIM: Control, Optimisation and Calculus of Variations (1998)

  • Volume: 3, page 1-22
  • ISSN: 1292-8119

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Mittenhuber, Dirk. "Dubins' problem is intrinsically three-dimensional." ESAIM: Control, Optimisation and Calculus of Variations 3 (1998): 1-22. <http://eudml.org/doc/90519>.

@article{Mittenhuber1998,
author = {Mittenhuber, Dirk},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal arcs; Pontryagin's maximum principle; Dubins problem},
language = {eng},
pages = {1-22},
publisher = {EDP Sciences},
title = {Dubins' problem is intrinsically three-dimensional},
url = {http://eudml.org/doc/90519},
volume = {3},
year = {1998},
}

TY - JOUR
AU - Mittenhuber, Dirk
TI - Dubins' problem is intrinsically three-dimensional
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 1998
PB - EDP Sciences
VL - 3
SP - 1
EP - 22
LA - eng
KW - optimal arcs; Pontryagin's maximum principle; Dubins problem
UR - http://eudml.org/doc/90519
ER -

References

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  1. [1] L.E. Dubins: On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents, Am. J. Math., 79, 1957, 497-516. Zbl0098.35401MR89457
  2. [2] V. Guillemin, S. Sternberg: Symplectic techniques in physics, Cambridge University Press, 1984. Zbl0576.58012MR770935
  3. [3] V. Jurdjevic: Non Euclidean Elastica, Am. J. Math., 117, 1995, 93-124. Zbl0822.58010MR1314459
  4. [4] V. Jurdjevic: Casimir elements and optimal control, in Geometry in nonlinear control and differential inclusions, B. Jakubczyk ed., Polish Academy of Sciences, Warsaw, Banach Center Publ, 32, 1995, 261-275. Zbl0859.49022MR1364433
  5. [5] V. Jurdjevic: Geometric Control Theory, Cambridge University Press, 1997. Zbl0940.93005MR1425878
  6. [6] A.J. Krener, H. Schättler: The structure of small-time reachable sets in low dimensions, SIAM J. Control Optimization, 27, 1989, 120-147. Zbl0669.49020MR980227
  7. [7] I.A.K. Kupka: Geometric theory of extremals in optimal control problems. I: The Fold and the Maxwell Case, Transactions of the AMS, 299, 1987, 225-243. Zbl0606.49016MR869409
  8. [8] I.A.K. Kupka: The ubiquity of Fuller's phenomenon, in Non-linear controllability and optimal control, H. J. Sussmann ed., Marcel Dekker, New York, 1990, 313-350. Zbl0739.49001MR1061391
  9. [9] J.E. Marsden, T. Ratiu: Introduction to Mechanics and Symmetry, Springer, New York, 1994. Zbl0811.70002MR1304682
  10. [10] D. Mittenhuber: Dubins' problem in H3, integration of abnormal extremals, in preparation. 
  11. [11] D. Mittenhuber: Dubins' problem in hyperbolic space, in Control Theory and its Applications, R. Sharpe ed., Can. Math. Soc. Conf. Proc. Ser., to appear. Zbl0980.53052MR1648712
  12. [12] D. Mittenhuber: Dubins' problem in the hyperbolic plane using the open disc model, in Control Theory and its Applications, R. Sharpe ed., Can. Math. Soc. Conf. Proc. Ser., to appear. Zbl0980.53053MR1648713
  13. [13] D. Mittenhuber: Applications of the Maximum Principle to Problems in Lie Semigroups, in Semigroups in Algebra, Geometry, and Analysis, K. H. Hofmann, J. D. Lawson and E. B. Vinberg. eds., de Gruyter, Berlin, 1995, 311-336. Zbl0898.60092MR1350338
  14. [14] F. Monroy Pérez: Non-Euclidean Dubins' problem, in Control Theory and its Applications, R. Sharpe ed., Can. Math. Soc. Conf. Proc. Ser., to appear. MR1626545
  15. [15] F. Monroy Pérez: Non-Euclidean Dubins'problem: A control theoretic approach, PhD thesis, University of Toronto, 1995. 
  16. [16] J.G. Ratcliffe: Foundations of Hyperbolic Manifolds, Springer, Berlin, New York, 1994. Zbl0809.51001MR1299730
  17. [17] J.A. Reeds, L.A. Shepp: Optimal paths for a car that goes both forwards and backwards, Pacific Journal of Mathematics, 145, 1990, 367-393. MR1069892
  18. [18] H.J. Sussmann: Shortest 3-dimensional paths with a prescribed curvature bound, in Proceedings of the 1995 IEEE Conference on Decision and Control, 1995, 3306-3312. 

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