Strong minimality of abnormal geodesics for \(2\)-distributions. (English) Zbl 0951.53029

Agrachëv, A. A.; Sarychev, A. V.

J. Dyn. Control Syst. 1, No. 2, 139-176 (1995).

Let \((M,g)\) be a Riemannian manifold, and let \(\mathcal D\) be a rank \(k\) distribution in \(TM\), such that iterated Lie brackets of vector fields in \(\mathcal D\) generate \(TM\). Locally Lipschitzian curves whose tangent vector belongs to \(\mathcal D\) almost everywhere are called admissible paths of \((M,\mathcal D)\). In contrast to the Riemannian setting, the space of admissible paths between two points \(p\), \(q\in M\) does not form a Banach manifold but may have singularities, called abnormal geodesics; these abnormal geodesics depend only on \(\mathcal D\), not on the metric \(g\). For example, certain admissible paths between \(p\) and \(q\) can be rigid, i.e., isolated in \(W_{1,\infty}\)-topology.
The authors give sufficient conditions for abnormal geodesics to be strongly (\(W_{1,1}\)-locally) minimal when \(\mathcal D\) has rank 2. The most important of these conditions is the strong generalized Legendre condition, also known as generalized Legendre-Clebsch condition or Kelley-condition [H. J. Kelly, R. E. Kopp and G. H. Moyer, in: G. Leitmann (ed.), “Topics in optimization”, Mathematics in Science and Engineering. 31 (1967; Zbl 0199.48602)]. The relation between strong minimality and rigidity is also discussed, cf. the paper by the authors in [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 13, 635-690 (1996; Zbl 0866.58023)].

Reviewer: Sebastian Goette (Tübingen)

MSC:

53C22	Geodesics in global differential geometry
58E10	Variational problems in applications to the theory of geodesics (problems in one independent variable)
49Q20	Variational problems in a geometric measure-theoretic setting
53C17	Sub-Riemannian geometry

Keywords:

sub-Riemannian geodesic; abnormal geodesic; strong generalized Legendre condition

Citations:

Zbl 0199.48602; Zbl 0866.58023

References:

[1]	A. A. Agrachev, Quadratic mappings in geometric control theory. (Russian)Itogi Nauki i Tekhniki; Problemy Geometrii, VINITI, Akad. Nauk SSSR, Vol. 20Moscow, 1988, 11–205. English translation:J. Sov. Math. 51 (1990), 2667–2734.
[2]	A. A. Agrachev and R.V. Gamkrelidze, Second-order optimality condition for the time-optimal problem. (Russian)Mat. Sb. 100 (1976), 610–643. English translation:Math. USSR Sb. 29 (1976), 547–576.
[3]	–, Exponential representation of flows and chronological calculus. (Russian)Mat. Sb. 107 (1978), 467–532. English translation:Math. USSR Sb.,35 (1979), 727–785. · Zbl 0408.34044
[4]	A. A. Agrachev, R. V. Gamkrelidze, and A. V. Sarychev, Local invariants of smooth control systems.Acta Appl. Math. 14 (1989), 191–237. · Zbl 0681.49018
[5]	A. A. Agrachev and A. V. Sarychev On abnormal extremals for lagrange variational problems.J. Math. Syst., Estimation and Control (to appear). · Zbl 0826.49012
[6]	A. A. Agrachev, Abnormal sub-Riemannian geodesics: Morse index and rigidity. Submitted toAnn. Inst. H. Poincaré, Anal. Nonlinéaire’. · Zbl 0866.58023
[7]	R. Bryant and L. Hsu, Rigidity of integral curves of rank 2 distributions.Invent. Math. 114 (1993), 435–461. · Zbl 0807.58007
[8]	V. Gershkovich, Engel structures on four-dimensional manifolds.Univ. Melbourne, Depart. Math., Preprint Series, 1992, No. 10.
[9]	V. Guillemin and S. Sternberg, Geometric asymptotics.Am. Math. Soc., Providence, Rhode Island, 1977.
[10]	M. Hestenes, Application of the theory of quadratic forms in Hilbert space to the calculus of variations.Pac. J. Math. 1 (1951), 525–582. · Zbl 0045.20806
[11]	H. J. Kelley R. Kopp, and H. G. Moyer, Singular extremals. In G. Leitmann, ed., Topics in Optimization.Academic Press, New York, 1967, 63–101.
[12]	A. J. Krener, The high-order maximum principle and its applications to singular extremals.SIAM J. Control Optim. 15 (1977), 256–293. · Zbl 0354.49008
[13]	I. Kupka Abnormal extremals.Preprint, 1992.
[14]	C. Lobry, Dynamical polysystems and control theory. In D. Q. Mayne and R. W. Brockett, eds., Geometric Methods in Systems Theory,Reidel, Dordrecht-Boston, 1973, 1–42. · Zbl 0279.93012
[15]	R. Montgomery, Abnormal minimizers,SIAM J. Control 32 (1994), 1605–1620. · Zbl 0816.49019
[16]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko, The mathematical theory of optimal processes.Pergamon Press, Oxford, 1964.
[17]	A. V. Sarychev, The index of the second variation of a control system. (Russian)Mat. Sb. 113 (1980), 464–486. English translation:Math. USSR Sb. 41 (1982), 383–401.
[18]	A. V. Sarychev, On Legendre-Jacobi-Morse-type theory of second variation for optimal control problems. Schwerpunktprogramm ”Anwendungsbezogene Optimierung und Steuerung” der Deutschen Forschungsgemeinschaft.Würzburg, 1992, Report No. 382.
[19]	H. J. Sussmann, A cornucopia of abnormal sub-Riemannian minimizers. Part I. The four-dimensional caseIMA Preprint Series, 1992, No. 1073.
[20]	H. J. Sussmann, Wensheng Liu, Shortest paths for sub-Riemannian metrics on rank 2 distributions.Rutgers Center for System and Control, 1993, Report SYCON-93-08.