Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity
ESAIM: Control, Optimisation and Calculus of Variations, Volume 4 (1999), pp. 377-403.
@article{COCV_1999__4__377_0,
     author = {Grachev, Andrei A. and Sarychev, Andrei V.},
     title = {Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {377--403},
     publisher = {EDP-Sciences},
     volume = {4},
     year = {1999},
     zbl = {0978.53065},
     mrnumber = {1693912},
     language = {en},
     url = {http://www.numdam.org/item/COCV_1999__4__377_0/}
}
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
SP  - 377
EP  - 403
VL  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/item/COCV_1999__4__377_0/
LA  - en
ID  - COCV_1999__4__377_0
ER  - 
%0 Journal Article
%A Grachev, Andrei A.
%A Sarychev, Andrei V.
%T Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 1999
%P 377-403
%V 4
%I EDP-Sciences
%U http://www.numdam.org/item/COCV_1999__4__377_0/
%G en
%F COCV_1999__4__377_0
Grachev, Andrei A.; Sarychev, Andrei V. Sub-riemannian metrics : minimality of abnormal geodesics versus subanalyticity. ESAIM: Control, Optimisation and Calculus of Variations, Volume 4 (1999), pp. 377-403. http://www.numdam.org/item/COCV_1999__4__377_0/

[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. | MR | Zbl

[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). | MR | Zbl

[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. | MR | Zbl

[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. | EuDML | Numdam | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[7] A.A. Agrachev, R. V. Gamkrelidze and A. V. Sarychev, Local invariants of smooth control systems. Acta Appl. Math. 14 ( 1989) 191-237. | MR | Zbl

[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. | MR | Zbl

[9] A.A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré 13 ( 1996) 635-690. | EuDML | Numdam | MR | Zbl

[10] A.A. Agrachev and A. V. Sarychev, Strong minimality of abnormal geodesics for 2-distributions. J. Dynamical Control Systems 1 ( 1995) 139-176. | MR | Zbl

[11] V.I. Arnol'D, A.N. Varchenko and S.M. Gusein-Zade, Singularities of differentiable maps 1 Birkhäuser, Boston ( 1985). | MR

[12] P. Brunovsky, Existence of regular synthesis for general problems. J. Differential Equations 38 ( 1980) 317-343. | MR | Zbl

[13] R.L. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 ( 1993) 435-461. | MR | Zbl

[14] W-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster ordnung, Match. Ann. 117, ( 1940/41) 98-105. | JFM

[15] A.F. Filippov, On certain questions in the theory of optimal control. Vestnik Moskov. Univ., Ser. Matem., Mekhan., Astron. 2 ( 1959) 25-32. | Zbl

[16] A. Gabrielov, Projections of semianalytic sets. Funct. Anal Appl. 2 ( 1968) 282-291. | MR | Zbl

[17] R. V. Gamkrelidze, Principles of optimal control theory. Plenum Press, New York ( 1978). | MR | Zbl

[18] Zhong Ge, Horizontal path space and Carnot-Caratheodory metric. Pacific J. Math. 161 ( 1993) 255-286. | MR | Zbl

[19] V. Ya. Gershkovich, Bilateral estimates for metrics, generated by completely nonholonomic distributions on Riemannian manifolds. Doklady AN SSSR 278 ( 1984) 1040-1044. | MR | Zbl

[20] B.S. Goh, Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 ( 1966) 716-731. | MR | Zbl

[21] M. Goresky and R. Macpherson, Stratified Morse Theory. Springer-Verlag, N.Y. ( 1988) Ch. 1. | MR | Zbl

[22] R. Hardt, Stratifications of real analytic maps and images. Inventiones Math. 28 ( 1975) 193-208. | MR | Zbl

[23] G. W. Haynes and H. Hermes, Nonlinear Controllability via Lie Theory. SIAM J. Control 8 ( 1970) 450-460. | MR | Zbl

[24] H. Hironaka, Subanalytic sets, Lecture Notes Istituto Matematico "Leonida Tonelli", Pisa, Italy ( 1973). | MR

[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. | MR

[26] A.J. Krener, The high-order maximum principle and its applications to singular extremals. SIAM J. Control and Optim. 15 ( 1977) 256-293. | MR | Zbl

[27] W. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-2 distributions, Memoirs of AMS, No. 564 ( 1995). | Zbl

[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. | MR | Zbl

[29] R. Montgomery, Geodesics, which do not satisfy geodesie equations, Preprint ( 1991).

[30] R. Montgomery, A survey on singular curves in sub-Riemannian geometry. J. Dynamical and Control Systems 1 ( 1995) 49-90. | MR | Zbl

[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] 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. | MR | Zbl

[33] J.P. Serre, Lie algebras and lie groups, Benjamin, New York ( 1965). | MR | Zbl

[34] H.J. Sussmann, Subanalytic sets and feedback control. J. Differential Equations 31 ( 1979) 31-52. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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). | Zbl

[38] L.C. Young, Lectures on the calculus of variations and optimal control theory, Chelsea, New York ( 1980).