Geodesic
Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group
Sbornik. Mathematics, Tome 202 (2011) no. 11, pp. 1593-1615 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

On the Engel group a nilpotent sub-Riemannian problem is considered, a 4-dimensional optimal control problem with a 2-dimensional linear control and an integral cost functional. It arises as a nilpotent approximation to nonholonomic systems with 2-dimensional control in a 4-dimensional space (for example, a system describing the navigation of a mobile robot with trailer). A parametrization of extremal trajectories by Jacobi functions is obtained. A discrete symmetry group and its fixed points, which are Maxwell points, are described. An estimate for the cut time (the time of the loss of optimality) on extremal trajectories is derived on this basis. Bibliography: 25 titles.
Keywords: optimal control, sub-Riemannian geometry, geometric methods, Engel group. 
@article{SM_2011_202_11_a1,
     author = {A. A. Ardentov and Yu. L. Sachkov},
     title = {Extremal trajectories in a~nilpotent {sub-Riemannian} problem on the {Engel} group},
     journal = {Sbornik. Mathematics},
     pages = {1593--1615},
     year = {2011},
     volume = {202},
     number = {11},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2011_202_11_a1/}
}
TY  - JOUR
AU  - A. A. Ardentov
AU  - Yu. L. Sachkov
TI  - Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group
JO  - Sbornik. Mathematics
PY  - 2011
SP  - 1593
EP  - 1615
VL  - 202
IS  - 11
UR  - http://geodesic.mathdoc.fr/item/SM_2011_202_11_a1/
LA  - en
ID  - SM_2011_202_11_a1
ER  - 
%0 Journal Article
%A A. A. Ardentov
%A Yu. L. Sachkov
%T Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group
%J Sbornik. Mathematics
%D 2011
%P 1593-1615
%V 202
%N 11
%U http://geodesic.mathdoc.fr/item/SM_2011_202_11_a1/
%G en
%F SM_2011_202_11_a1
A. A. Ardentov; Yu. L. Sachkov. Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group. Sbornik. Mathematics, Tome 202 (2011) no. 11, pp. 1593-1615. http://geodesic.mathdoc.fr/item/SM_2011_202_11_a1/

[1] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., 91, Amer. Math. Soc., Providence, RI, 2002 | MR | Zbl

[2] A. Bellaiche, “The tangent space in sub-Riemannian geometry”, Sub-Riemannian geometry (Paris, France, 1992), Progr. Math., 144, Birkhäuser, Basel, 1996, 1–78 | MR | Zbl

[3] A. A. Agrachev, A. V. Sarychev, “Filtrations of a Lie algebra of vector fields and nilpoint approximation of controlled systems”, Soviet Math. Dokl., 36:1 (1988), 104–108 | MR | Zbl

[4] H. Hermes, “Nilpotent and high-order approximations of vector field systems”, SIAM Rev., 33:2 (1991), 238–264 | DOI | MR | Zbl

[5] A. M. Vershik, V. Ya. Gershkovich, “Nonholonomic dynamical systems, geometry of distributions and variational problems”, Dynamical systems. VII, Encycl. Math. Sci., 16, 1994, 1–81 | MR | Zbl

[6] U. Boscain, F. Rossi, “Invariant Carnot–Caratheodory metrics on $S^3$, $\operatorname{SO}(3)$, $\operatorname{SL}(2)$ and lens spaces”, SIAM J. Control Optim., 47:4 (2008), 1851–1878 | DOI | MR | Zbl

[7] O. Myasnichenko, “Nilpotent $(3,6)$ sub-Riemannian problem”, J. Dynam. Control Systems, 8:4 (2002), 573–597 | DOI | MR | Zbl

[8] O. Myasnichenko, “Nilpotent $(n,n(n+1)/2)$ sub-Riemannian problem”, J. Dyn. Control Syst., 12:1 (2006), 87–95 | DOI | MR | Zbl

[9] Yu. L. Sachkov, “Complete description of the Maxwell strata in the generalized Dido problem”, Sb. Math., 197:6 (2006), 901–950 | DOI | MR | Zbl

[10] Yu. L. Sachkov, “Maxwell strata and symmetries in the problem of optimal rolling of a sphere over a plane”, Sb. Math., 201:7 (2010), 1029–1051 | DOI | Zbl

[11] F. Monroy-Pérez, A. Anzaldo-Meneses, “The step-2 nilpotent $(n,n(n+1)/2)$ sub-Riemannian geometry”, J. Dyn. Control Syst., 12:2 (2006), 185–216 | DOI | MR | Zbl

[12] I. Moiseev, Yu. L. Sachkov, “Maxwell strata in sub-Riemannian problem on the group of motions of a plane”, ESAIM Control Optim. Calc. Var., 16:2 (2010), 380–399 | DOI | MR | Zbl

[13] A. A. Agrachev, “Exponential mappings for contact sub-Riemannian structures”, J. Dynam. Control Systems, 2:3 (1996), 321–358 | DOI | MR | Zbl

[14] A. Agrachev, B. Bonnard, M. Chyba, I. Kupka, “Sub-Riemannian sphere in Martinet flat case”, ESAIM Control Optim. Calc. Var., 2 (1997), 377–448 | DOI | MR | Zbl

[15] A. A. Agrachev, Yu. L. Sachkov, Geometricheskaya teoriya upravleniya, Fizmatlit, M., 2005 | Zbl

[16] J. P. Laumond, Robot motion planning and control, Lecture Notes in Control and Inform. Sci., 229, Springer-Verlag, Berlin, 1998 | MR

[17] Yu. L. Sachkov, “Maxwell strata in the Euler elastic problem”, J. Dyn. Control Syst., 14:2 (2008), 169–234 | DOI | MR | Zbl

[18] Yu. L. Sachkov, “Conjugate points in Euler's elastic problem”, J. Dyn. Control Syst., 14:3 (2008), 409–439 | DOI | MR | Zbl

[19] Yu. L. Sachkov, Upravlyaemost i simmetrii invariantnykh sistem na gruppakh Li i odnorodnykh prostranstvakh, Fizmatlit, M., 2007 | Zbl

[20] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, Selected works. Vol. 4. The mathematical theory of optimal processes, Classics Soviet Math., Gordon Breach, New York, 1986 | MR | MR | Zbl | Zbl

[21] L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici latissimo sensu accepti, Universitat Autónoma de Barcelona, Barcelona, 1993 | MR | Zbl

[22] A. E. H. Love, A treatise on the mathematical theory of elasticity, Cambridge Univ. Press, Cambridge, 1927 | MR | Zbl

[23] V. Jurdjevic, Geometric control theory, Cambridge Stud. Adv. Math., 52, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[24] E. T. Whittaker, G. N. Watson, A course of modern analysis, Cambridge Univ. Press, New York, 1962 | MR | Zbl | Zbl

[25] A. A. Ardentov, Yu. L. Sachkov, “Solution to Euler's elastic problem”, Autom. Remote Control, 70:4 (2009), 633–643 | DOI | MR | Zbl