Cut time in sub-riemannian problem on engel group
ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 958-988.

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. The global optimality of extremal trajectories is studied via geometric control theory. The global diffeomorphic structure of the exponential mapping is described. As a consequence, the cut time is proved to be equal to the first Maxwell time corresponding to discrete symmetries of the exponential mapping.

Received:
DOI: 10.1051/cocv/2015027
Classification: 22E25, 58E25
Keywords: Sub-Riemannian geometry, optimal control, Engel group, Lie algebra, Maxwell time, cut time, exponential mapping, Euler’s elastica
Ardentov, A.A. 1; Sachkov, Yu.L. 1

1 Program Systems Institute of RAS, Pereslavl-Zalessky 152020, Russia
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Ardentov, A.A.; Sachkov, Yu.L. Cut time in sub-riemannian problem on engel group. ESAIM: Control, Optimisation and Calculus of Variations, Volume 21 (2015) no. 4, pp. 958-988. doi : 10.1051/cocv/2015027. http://www.numdam.org/articles/10.1051/cocv/2015027/

A.A. Agrachev, Geometry of optimal control problems and Hamiltonian systems. In: Nonlinear and Optimal Control Theory, Lect. Notes Math. CIME, 1932. Springer Verlag (2008) 1–59. | MR | Zbl

A.A. Agrachev and D. Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst. 18 (2012) 21–44. | DOI | MR | Zbl

A.A. Agrachev and Yu.L. Sachkov, Control Theory from the Geometric Viewpoint. Vol. 87 of Encycl. Math. Sci. Springer-Verlag (2004). | MR | Zbl

A.A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and sub-Riemannian geometry. Lect. Notes. Preprint (2014). Available at https://www.imj-prg.fr/˜davide.barilari/Notes.php.

A.A. Agrachev, B. Bonnard, M. Chyba and I. Kupka, Sub-Riemannian sphere in Martinet flat case. ESAIM: COCV 2 (1997) 377–448. | Numdam | MR | Zbl

A.A. Ardentov and Yu.L. Sachkov, Extremal trajectories in nilpotent sub-Riemannian problem on the Engel group. Sbornik: Math. 202 (2011) 1593–1615. | DOI | MR | Zbl

A.A. Ardentov and Yu.L. Sachkov, Conjugate points in nilpotent sub-Riemannian problem on the Engel group. J. Math. Sci. 195 (2013) 369–390. | DOI | MR | Zbl

D.M. Almeida, Sub-Riemannian homogeneous spaces of Engel type. J. Dyn. Control Syst. 20 (2014) 149–166. | DOI | MR | Zbl

V.N. Berestovskii and I.A. Zubareva, Shapes of spheres of special nonholonomic left-invariant intrinsic metrics on some Lie groups, Siber. Math. J. 42 (2001) 613–628. | DOI | MR | Zbl

B. Bonnard and M. Chyba, Singular Trajectories and their Role in Control Theory. Springer (2003). | MR | Zbl

U. Boscain and F. Rossi, Invariant Carnot-Caratheodory metrics on S 3 , SO(3), SL(2) and Lens spaces. SIAM J. Control Optim. 47 (2008) 1851–1878. | DOI | MR | Zbl

Y.A. Butt, Yu.L. Sachkov and A.I. Bhatti, Parametrization of extremal trajectories in sub-Riemannian problem on group of motions of pseudo euclidean plane. J. Dyn. Control Syst. 20 (2014) 341–364. | DOI | MR | Zbl

Y.A. Butt, Yu.L. Sachkov and A.I. Bhatti, Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2). Preprint arXiv:1408.2043v1 (2014). | MR

M. Christ, Nonexistence of invariant analytic hypoelliptic differential operators on nilpotent groups of step greater than two. Essays on Fourier analysis in honor of Elias M. Stein, Vol. 42 of Princeton Math. Ser. (1995) 127–145. | MR | Zbl

L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive Solutio problematis isoperimitrici latissimo sensu accepti. Lausanne, Geneva (1744). | Zbl

J.P. Gauthier and V. Zakalyukin, On the one-step bracket-generating motion planning problem. J. Dyn. Control Syst. 11 (2005) 215–235. | DOI | MR | Zbl

J.P. Gauthier and V. Zakalyukin, On the motion planning problem, complexity, entropy and nonholonomic interpolation. J. Dyn. Control Syst. 12 (2006) 371–404. | DOI | MR | Zbl

V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997). | MR | Zbl

S.G. Krantz and H.R. Parks, The Implicit Function Theorem: History, Theory, and Applications. Birkauser (2001). | MR | Zbl

A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity. New York, Dover (1927). | JFM | MR | Zbl

I. Moiseev and Yu.L. Sachkov, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380–399. | Numdam | MR | Zbl

R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Vol. 94. Math. Surv. Monogr. AMS (2002). | MR | Zbl

L.S. Pontryagin, V.G. Boltayanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. John Wiley, New York, London (1962). | MR | Zbl

Yu.L. Sachkov, Exponential map in the generalized Dido problem. Sb. Math. 194 (2003) 1331–1359. | DOI | MR | Zbl

Yu.L. Sachkov, Discrete symmetries in the generalized Dido problem. Sb. Math. 197 (2006) 235–257. | DOI | MR | Zbl

Yu.L. Sachkov, The Maxwell set in the generalized Dido problem. Sb. Math. 197 (2006) 595–621. | DOI | MR | Zbl

Yu.L. Sachkov, Complete description of the Maxwell strata in the generalized Dido problem. Sb. Math. 197 (2006) 901–950. | DOI | MR | Zbl

Yu.L. Sachkov, Maxwell strata in Euler’s elastic problem. J. Dyn. Control Syst. 14 (2008) 169–234. | DOI | MR | Zbl

Yu.L. Sachkov, Conjugate points in Euler’s elastic problem. J. Dyn. Control Syst. Springer, New York (2008) 14 409–439. | MR | Zbl

Yu.L. Sachkov, Conjugate and cut time in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018–1039. | Numdam | MR | Zbl

Yu.L. Sachkov, Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 17 (2011) 293–321. | Numdam | MR | Zbl

Yu. L. Sachkov and E.F. Sachkova, Exponential mapping in Euler’s elastic problem. J. Dyn. Control Syst. 20 (2014) 443–464. | DOI | MR | Zbl

F. Trèves, Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the ¯-Neumann problem. Commun. Partial Differ. Equ. 3 (1978) 475–642. | DOI | MR | Zbl

A.M. Vershik and V.Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems. Dynamical Systems VII. Encycl. Math. Sci. (1990) 4–79. | Zbl

E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. Cambridge University Press (1927). | JFM | MR

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