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Sub-Riemannian Metrics: Minimality of Abnormal Geodesics versus Subanalyticity

Published online by Cambridge University Press:  15 August 2002

Andrei A. Agrachev  and
Andrei V. Sarychev
Andrei A. Agrachev
Affiliation:
Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina ul. 8, 117966 Moscow, Russia; [email protected]. SISSA, via Beirut 2-4, 34014 Trieste, Italy; [email protected].
Andrei V. Sarychev
Affiliation:
Department of Mathematics, University of Aveiro 3810-193 Aveiro, Portugal; [email protected].
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Abstract

We study sub-Riemannian (Carnot-Caratheodory) metrics defined by noninvolutive distributions on real-analytic Riemannian manifolds. We establish a connection between regularity properties of these metrics and the lack of length minimizing abnormal geodesics. Utilizing the results of the previous study of abnormal length minimizers accomplished by the authors in [Annales IHP. Analyse nonlinéaire 13, p. 635-690] we describe in this paper two classes of the germs of distributions (called 2-generating and medium fat) such that the corresponding sub-Riemannian metrics are subanalytic. To characterize these classes of distributions we determine the dimensions of the manifolds on which generic germs of distributions of given rank are respectively 2-generating or medium fat.

Keywords

Sub-Riemannian metricssubanalitycityabnormal length minimizers.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 377 - 403
Copyright
© EDP Sciences, SMAI, 1999

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References

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.
Agrachev, A.A., The second-order optimality condition in the general nonlinear case. Matem. Sbornik 102 (1977) 551-568. English transl. in: Math. USSR Sbornik 31 (1977).
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.
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.
Agrachev, A.A. and Gamkrelidze, R.V., Second-order optimality condition for the time-optimal problem. Matem. Sbornik 100 (1976) 610-643. English transl. in: Math. USSR Sbornik 29 (1976) 547-576.
Agrachev, A.A. and Gamkrelidze, R.V., Exponential representation of flows and chronological calculus. Matem. Sbornik 107 (1978) 467-532. English transl. in: Math. USSR Sbornik 35 (1979) 727-785.
Agrachev, A.A., Gamkrelidze, R.V. and Sarychev, A.V., Local invariants of smooth control systems. Acta Appl. Math. 14 (1989) 191-237. CrossRef
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.
Agrachev, A.A. and Sarychev, A.V., Abnormal sub-Riemannian geodesics: Morse index and rigidity. Ann. Inst. H. Poincaré 13 (1996) 635-690. CrossRef
Agrachev, A.A. and Sarychev, A.V., Strong minimality of abnormal geodesics for 2-distributions. J. Dynamical Control Systems 1 (1995) 139-176. CrossRef
V.I. Arnol'd, A.N. Varchenko and S.M. Gusein-Zade, Singularities of differentiable maps 1 Birkhäuser, Boston (1985).
Brunovsky, P., Existence of regular synthesis for general problems. J. Differential Equations 38 (1980) 317-343. CrossRef
Bryant, R.L. and Hsu, L., Rigidity of integral curves of rank 2 distributions. Invent. Math. 114 (1993) 435-461. CrossRef
W-L. Chow, Über Systeme von linearen partiellen Differentialgleichungen erster ordnung. Match. Ann. 117 , (1940/41) 98-105.
A.F. Filippov, On certain questions in the theory of optimal control. Vestnik Moskov. Univ., Ser. Matem., Mekhan., Astron. 2 (1959) 25-32.
Gabrielov, A., Projections of semianalytic sets. Funct. Anal Appl. 2 (1968) 282-291. CrossRef
R.V. Gamkrelidze, Principles of optimal control theory. Plenum Press, New York (1978).
Zhong, Ge, Horizontal path space and Carnot-Caratheodory metric. Pacific J. Math. 161 (1993) 255-286.
Gershkovich, V.Ya., Bilateral estimates for metrics, generated by completely nonholonomic distributions on Riemannian manifolds. Doklady AN SSSR 278 (1984) 1040-1044.
Goh, B.S., Necessary conditions for singular extremals involving multiple control variables. SIAM J. Control 4 (1966) 716-731. CrossRef
M. Goresky and R. MacPherson, Stratified Morse Theory. Springer-Verlag, N.Y. (1988) Ch.1.
Hardt, R., Stratifications of real analytic maps and images. Inventiones Math. 28 (1975) 193-208. CrossRef
Haynes, G.W. and Hermes, H., Nonlinear Controllability via Lie Theory. SIAM J. Control 8 (1970) 450-460. CrossRef
H. Hironaka, Subanalytic sets, Lecture Notes Istituto Matematico ``Leonida Tonelli'', Pisa, Italy (1973).
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.
Krener, A.J., The high-order maximum principle and its applications to singular extremals. SIAM J. Control and Optim. 15 (1977) 256-293. CrossRef
W. Liu and H.J. Sussmann, Shortest paths for sub-Riemannian metrics on rank-2 distributions, Memoirs of AMS, No. 564 (1995).
Lojasiewicz Jr, S.. 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. CrossRef
R. Montgomery, Geodesics, which do not satisfy geodesic equations, Preprint (1991).
Montgomery, R., A survey on singular curves in sub-Riemannian geometry. J. Dynamical and Control Systems 1 (1995) 49-90. CrossRef
Rashevsky, P.K., About connecting two points of a completely nonholonomic space by admissible curve. Uchen. Zap. Ped. Inst. Libknechta 2 (1938) 83-94.
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.
J.P. Serre, Lie algebras and lie groups, Benjamin, New York (1965).
Sussmann, H.J., Subanalytic sets and feedback control. J. Differential Equations 31 (1979) 31-52. CrossRef
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.
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.
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).
L.C. Young, Lectures on the calculus of variations and optimal control theory, Chelsea, New York (1980).