Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity

Andrei A. Grachev; Andrei V. Sarychev

ESAIM: Control, Optimisation and Calculus of Variations (1999)

  • Volume: 4, page 377-403
  • ISSN: 1292-8119

How to cite

top

Grachev, Andrei A., and Sarychev, Andrei V.. "Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity." ESAIM: Control, Optimisation and Calculus of Variations 4 (1999): 377-403. <http://eudml.org/doc/90545>.

@article{Grachev1999,
author = {Grachev, Andrei A., Sarychev, Andrei V.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {abnormal length minimizer; Carnot-Carathéodory metrics; noninvolutive distributions; abnormal geodesics; sub-Riemannian metrics; subanalytic},
language = {eng},
pages = {377-403},
publisher = {EDP Sciences},
title = {Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity},
url = {http://eudml.org/doc/90545},
volume = {4},
year = {1999},
}

TY - JOUR
AU - Grachev, Andrei A.
AU - Sarychev, Andrei V.
TI - Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 1999
PB - EDP Sciences
VL - 4
SP - 377
EP - 403
LA - eng
KW - abnormal length minimizer; Carnot-Carathéodory metrics; noninvolutive distributions; abnormal geodesics; sub-Riemannian metrics; subanalytic
UR - http://eudml.org/doc/90545
ER -

References

top
  1. [1] A.A. Agrachev, Quadratic mappings in geometric control theory, in: Itogi Nauki i Tekhniki, Problemy Geometrii, VINITI, Acad. Nauk SSSR, Moscow 20 ( 1988) 11-205. English transl. in J. Soviet Math. 51 ( 1990) 2667-2734. MR966201
  2. [2] A.A. Agrachev, The second-order optimality condition in the general nonlinear case. Matem. Sbornik 102 ( 1977) 551-568. English transl. in: Math. USSR Sbornik 31 ( 1977). Zbl0362.49016MR451084
  3. [3] A.A. Agrachev, Topology of quadratic mappings and Hessians of smooth mappings, in: Itogi Nauki i Tekhniki, Algebra, Topologia, Geometria; VINITI, Acad. Nauk SSSR 26 ( 1988) 85-124. Zbl0719.58006MR978394
  4. [4] A.A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, Sub-Riemannian spheres in Martinet flat case. ESAIM: Contr., Optim. and Calc. Var. 2 ( 1997) 377-448. Zbl0902.53033MR1483765
  5. [5] A.A. Agrachev and R.V. Gamkrelidze, Second-order optimality condition for the time-optimal problemMatem. Sbornik 100 ( 1976) 610-643. English transl. in: Math. USSR Sbornik 29 ( 1976) 547-576. Zbl0341.49007MR425719
  6. [6] A.A. Agrachev and R. V. Gamkrelidze, Exponential representation of flows and chronological calculus. Matem. Sbornik 107 ( 1978) 467-532. English transl. in: Math. USSR Sbornik 35 ( 1979) 727-785. Zbl0429.34044MR524203
  7. [7] A.A. Agrachev, R. V. Gamkrelidze and A. V. Sarychev, Local invariants of smooth control systems. Acta Appl. Math. 14 ( 1989) 191-237. Zbl0681.49018MR995286
  8. [8] A.A. Agrachev and A. V. Sarychev, On abnormal extremals for Lagrange variational problems. (summary). J. Mathematical Systems, Estimation and Control 5 ( 1995) 127-130. Complete version: J. Mathematical Systems, Estimation and Control 8 ( 1998) 87-118. Zbl0826.49012MR1486492
  9. [9] A.A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré 13 ( 1996) 635-690. Zbl0866.58023MR1420493
  10. [10] A.A. Agrachev and A. V. Sarychev, Strong minimality of abnormal geodesics for 2-distributions. J. Dynamical Control Systems 1 ( 1995) 139-176. Zbl0951.53029MR1333769
  11. [11] V.I. Arnol'd, A.N. Varchenko and S.M. Gusein-Zade, Singularities of differentiable maps 1 Birkhäuser, Boston ( 1985). MR777682
  12. [12] P. Brunovsky, Existence of regular synthesis for general problems. J. Differential Equations 38 ( 1980) 317-343. Zbl0417.49030MR605053
  13. [13] R.L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 ( 1993) 435-461. Zbl0807.58007MR1240644
  14. [14] W-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster ordnung, Match. Ann. 117, ( 1940/41) 98-105. Zbl65.0398.01JFM65.0398.01
  15. [15] A.F. Filippov, On certain questions in the theory of optimal control. Vestnik Moskov. Univ., Ser. Matem., Mekhan., Astron. 2 ( 1959) 25-32. Zbl0090.06902
  16. [16] A. Gabrielov, Projections of semianalytic sets. Funct. Anal Appl. 2 ( 1968) 282-291. Zbl0179.08503MR245831
  17. [17] R. V. Gamkrelidze, Principles of optimal control theory. Plenum Press, New York ( 1978). Zbl0401.49001MR686793
  18. [18] Zhong Ge, Horizontal path space and Carnot-Caratheodory metric. Pacific J. Math. 161 ( 1993) 255-286. Zbl0797.49033MR1242199
  19. [19] V. Ya. Gershkovich, Bilateral estimates for metrics, generated by completely nonholonomic distributions on Riemannian manifolds. Doklady AN SSSR 278 ( 1984) 1040-1044. Zbl0591.53033MR765608
  20. [20] B.S. Goh, Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 ( 1966) 716-731. Zbl0161.29004MR205719
  21. [21] M. Goresky and R. MacPherson, Stratified Morse Theory. Springer-Verlag, N.Y. ( 1988) Ch. 1. Zbl0639.14012MR932724
  22. [22] R. Hardt, Stratifications of real analytic maps and images. Inventiones Math. 28 ( 1975) 193-208. Zbl0298.32003MR372237
  23. [23] G. W. Haynes and H. Hermes, Nonlinear Controllability via Lie Theory. SIAM J. Control 8 ( 1970) 450-460. Zbl0229.93012MR277859
  24. [24] H. Hironaka, Subanalytic sets, Lecture Notes Istituto Matematico "Leonida Tonelli", Pisa, Italy ( 1973). MR377101
  25. [25] H.J. Kelley, R. Kopp and H.G. Moyer, Singular Extremals, G. Leitman, Ed., Topics in Optimization, Academic Press, New York, N.Y. ( 1967) 63-101. MR215153
  26. [26] A.J. Krener, The high-order maximum principle and its applications to singular extremals. SIAM J. Control and Optim. 15 ( 1977) 256-293. Zbl0354.49008MR433288
  27. [27] W. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-2 distributions, Memoirs of AMS, No. 564 ( 1995). Zbl0843.53038
  28. [28] S. Jr. Lojasiewicz and H.J. Sussmann, Some examples of reachable sets and optimal cost functions that fail to be subanalytic. SIAM J. Control and Optim. 23 ( 1985) 584-598. Zbl0569.49029MR791889
  29. [29] R. Montgomery, Geodesics, which do not satisfy geodesie equations, Preprint ( 1991). 
  30. [30] R. Montgomery, A survey on singular curves in sub-Riemannian geometry. J. Dynamical and Control Systems 1 ( 1995) 49-90. Zbl0941.53021MR1319057
  31. [31] P.K. Rashevsky, About connecting two points of a completely nonholonomic space by admissible curve. Uchen. Zap. Ped. Inst. Libknechta 2 ( 1938) 83-94. 
  32. [32] C.B. Rayner, The exponential map for the Lagrange problem on differentiable manifolds. Philos. Trans. Roy. Soc. London Ser. A, Math. Phys. Sci. 262 ( 1967) 299-344. Zbl0154.37004MR247552
  33. [33] J.P. Serre, Lie algebras and lie groups, Benjamin, New York ( 1965). Zbl0132.27803MR218496
  34. [34] H.J. Sussmann, Subanalytic sets and feedback control. J. Differential Equations 31 ( 1979) 31-52. Zbl0407.93010MR524816
  35. [35] H.J. Sussmann, A cornucopia of four-dimensional abnormall sub-Riemannian minimizers, A. Bellaïche, J.-J. Risler, Eds., Sub-Riemannian Geometry, Birkhäuser, Basel ( 1996) 341-364. Zbl0862.53033MR1421825
  36. [36] H.J. Sussmann, Optimal control and piecewise analyticity of the distance function. A. Ioffe, S. Reich, Eds., Pitman Research Notes in Mathematics, Longman Publishers ( 1992) 298-310. Zbl0772.49019MR1184651
  37. [37] A.M. Vershik and V.Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems. V.I. Arnol'd, S.P. Novikov, Eds., Dynamical systems VII, Encyclopedia of Mathematical Sciences 16, Springer-Verlag, NY ( 1994). Zbl0797.58007
  38. [38] L.C. Young, Lectures on the calculus of variations and optimal control theory, Chelsea, New York ( 1980). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.