@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}, mrnumber = {1693912}, zbl = {0978.53065}, 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, Tome 4 (1999), pp. 377-403. http://www.numdam.org/item/COCV_1999__4__377_0/
[1] 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] 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] 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] Sub-Riemannian spheres in Martinet flat case. ESAIM: Contr., Optim. and Calc. Var. 2 ( 1997) 377-448. | EuDML | Numdam | MR | Zbl
, , and ,[5] 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
and ,[6] 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
and ,[7] Local invariants of smooth control systems. Acta Appl. Math. 14 ( 1989) 191-237. | MR | Zbl
, and ,[8] 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
and ,[9] Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré 13 ( 1996) 635-690. | EuDML | Numdam | MR | Zbl
and ,[10] Strong minimality of abnormal geodesics for 2-distributions. J. Dynamical Control Systems 1 ( 1995) 139-176. | MR | Zbl
and ,[11] Singularities of differentiable maps 1 Birkhäuser, Boston ( 1985). | MR
, and ,[12] Existence of regular synthesis for general problems. J. Differential Equations 38 ( 1980) 317-343. | MR | Zbl
,[13] Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 ( 1993) 435-461. | MR | Zbl
and ,[14] Über Systeme von linearen partiellen Differentialgleichungen erster ordnung, Match. Ann. 117, ( 1940/41) 98-105. | JFM
,[15] On certain questions in the theory of optimal control. Vestnik Moskov. Univ., Ser. Matem., Mekhan., Astron. 2 ( 1959) 25-32. | Zbl
,[16] Projections of semianalytic sets. Funct. Anal Appl. 2 ( 1968) 282-291. | MR | Zbl
,[17] Principles of optimal control theory. Plenum Press, New York ( 1978). | MR | Zbl
,[18] Horizontal path space and Carnot-Caratheodory metric. Pacific J. Math. 161 ( 1993) 255-286. | MR | Zbl
,[19] Bilateral estimates for metrics, generated by completely nonholonomic distributions on Riemannian manifolds. Doklady AN SSSR 278 ( 1984) 1040-1044. | MR | Zbl
,[20] Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 ( 1966) 716-731. | MR | Zbl
,[21] Stratified Morse Theory. Springer-Verlag, N.Y. ( 1988) Ch. 1. | MR | Zbl
and ,[22] Stratifications of real analytic maps and images. Inventiones Math. 28 ( 1975) 193-208. | MR | Zbl
,[23] Nonlinear Controllability via Lie Theory. SIAM J. Control 8 ( 1970) 450-460. | MR | Zbl
and ,[24] Subanalytic sets, Lecture Notes Istituto Matematico "Leonida Tonelli", Pisa, Italy ( 1973). | MR
,[25] Singular Extremals, G. Leitman, Ed., Topics in Optimization, Academic Press, New York, N.Y. ( 1967) 63-101. | MR
, and ,[26] The high-order maximum principle and its applications to singular extremals. SIAM J. Control and Optim. 15 ( 1977) 256-293. | MR | Zbl
,[27] Shortest paths for sub-Riemannian metrics on rank-2 distributions, Memoirs of AMS, No. 564 ( 1995). | Zbl
and ,[28] 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
and ,[29] Geodesics, which do not satisfy geodesie equations, Preprint ( 1991).
,[30] A survey on singular curves in sub-Riemannian geometry. J. Dynamical and Control Systems 1 ( 1995) 49-90. | MR | Zbl
,[31] About connecting two points of a completely nonholonomic space by admissible curve. Uchen. Zap. Ped. Inst. Libknechta 2 ( 1938) 83-94.
,[32] 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] Lie algebras and lie groups, Benjamin, New York ( 1965). | MR | Zbl
,[34] Subanalytic sets and feedback control. J. Differential Equations 31 ( 1979) 31-52. | MR | Zbl
,[35] 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] 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] 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
and ,[38] Lectures on the calculus of variations and optimal control theory, Chelsea, New York ( 1980).
,