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Continuous dependence in hyperbolic problems with Wentzell boundary conditions
Geometric conditions for the existence of a rolling without twisting or slipping
1. | Mathematisches Institut, Georg-August-Universität, Bunsen-str. 3-5, D-37073 Göttingen, Germany |
2. | Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020 |
References:
[1] |
A. Agrachev, Rolling balls and octonions,, Proc. Steklov Inst. Math., 258 (2007), 13.
doi: 10.1134/S0081543807030030. |
[2] |
A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,'', Springer, (2004).
doi: 10.1007/978-3-662-06404-7. |
[3] |
A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics,, Notices Amer. Math. Soc., 52 (2005), 324.
|
[4] |
G. Bor and R. Montgomery, $G_2$ and the rolling distribution,, L'Ens. Math., 55 (2009), 157.
|
[5] |
É. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre,, Ann. Sci. \'Ecole Norm. Sup., 27 (1910), 109.
|
[6] |
S. A. Chaplygin, On some generalization of the area theorem, with applications to the problem of rolling balls, (Russian), Mat. Sbornik, XX (1897), 1.
doi: 10.1134/S1560354712020086. |
[7] |
S. A. Chaplygin, On a ball's rolling on a horizontal plane, (Russian), Mat. Sbornik, XXIV (1903), 139.
doi: 10.1070/RD2002v007n02ABEH000200. |
[8] |
Y. Chitour, A. Marigo and B. Piccoli, Quantization of the rolling-body problem with applications to motion planning,, Systems Control Lett., 54 (2005), 999.
doi: 10.1016/j.sysconle.2005.02.012. |
[9] |
Y. Chitour, M. Godoy Molina and P. Kokkonen, Symmetries of the rolling model, preprint,, \arXiv{1301.2579}., (). Google Scholar |
[10] |
Y. Chitour and P. Kokkonen, Rolling manifolds: Intrinsic formulation and controllability, preprint,, \arXiv{1011.2925}., (). Google Scholar |
[11] |
Y. Chitour and P. Kokkonen, Rolling manifolds on space forms,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 29 (2012), 927.
doi: 10.1016/j.anihpc.2012.05.005. |
[12] |
B. E. J. Dahlberg, The converse of the four vertex theorem,, Proc. Amer. Math. Soc., 133 (2005), 2131.
doi: 10.1090/S0002-9939-05-07788-9. |
[13] |
M. Godoy Molina, E. Grong, I. Markina and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds,, J. Dyn. Control Syst., 18 (2012), 181.
doi: 10.1007/s10883-012-9139-2. |
[14] |
E. Grong, Controllability of rolling without twisting or slipping in higher dimensions,, SIAM J. Control Optim., 50 (2012), 2462.
doi: 10.1137/110829581. |
[15] |
E. Hsu, "Stochastic Analysis on Manifolds,'' Graduate Studies in Mathematics 38,, American Mathematical Society, (2002).
|
[16] |
K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Grassmann manifolds,, J. Dyn. Control Syst., 13 (2007), 467.
doi: 10.1007/s10883-007-9027-3. |
[17] |
B. D. Johnson, The nonholonomy of the rolling sphere,, Amer. Math. Monthly, 114 (2007), 500.
|
[18] |
V. Jurdjevic and J. A. Zimmerman, Rolling sphere problems on spaces of constant curvature,, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729.
doi: 10.1017/S0305004108001084. |
[19] |
M. Levi, Geometric phases in the motion of rigid bodies,, Arch. Rational Mech. Anal., 122 (1993), 213.
doi: 10.1007/BF00380255. |
[20] |
K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact,, T\^ohoku Math. J., 30 (1978), 623.
doi: 10.2748/tmj/1178229921. |
[21] |
J. W. Robbin and D. A. Salamon, "Introduction to Differential Geometry,", Available from: \url{http://www.math.ethz.ch/\, (). Google Scholar |
[22] |
R. W. Sharpe, "Differential Geometry,'', GTM 166, (1997).
|
[23] |
M. Spivak, "A Comprehensive Introduction to Differential Geometry'',, Volume IV, (1999).
|
[24] |
I. Zelenko, On variational approach to differential invariants of rank two distributions,, Differential Geom. Appl., 24 (2006), 235.
doi: 10.1016/j.difgeo.2005.09.004. |
[25] |
I. Zelenko, Fundamental form and the Cartan tensor of $(2,5)$-distributions coincide,, J. Dyn. Control Syst., 12 (2006), 247.
doi: 10.1007/s10450-006-0383-1. |
[26] |
J. A. Zimmerman, Optimal control of the sphere $S^n$ rolling on $E^n$,, Math. Control Signals Systems, 17 (2005), 14.
doi: 10.1007/s00498-004-0143-2. |
show all references
References:
[1] |
A. Agrachev, Rolling balls and octonions,, Proc. Steklov Inst. Math., 258 (2007), 13.
doi: 10.1134/S0081543807030030. |
[2] |
A. Agrachev and Y. Sachkov, "Control Theory from the Geometric Viewpoint,'', Springer, (2004).
doi: 10.1007/978-3-662-06404-7. |
[3] |
A. M. Bloch, J. E. Marsden and D. V. Zenkov, Nonholonomic dynamics,, Notices Amer. Math. Soc., 52 (2005), 324.
|
[4] |
G. Bor and R. Montgomery, $G_2$ and the rolling distribution,, L'Ens. Math., 55 (2009), 157.
|
[5] |
É. Cartan, Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre,, Ann. Sci. \'Ecole Norm. Sup., 27 (1910), 109.
|
[6] |
S. A. Chaplygin, On some generalization of the area theorem, with applications to the problem of rolling balls, (Russian), Mat. Sbornik, XX (1897), 1.
doi: 10.1134/S1560354712020086. |
[7] |
S. A. Chaplygin, On a ball's rolling on a horizontal plane, (Russian), Mat. Sbornik, XXIV (1903), 139.
doi: 10.1070/RD2002v007n02ABEH000200. |
[8] |
Y. Chitour, A. Marigo and B. Piccoli, Quantization of the rolling-body problem with applications to motion planning,, Systems Control Lett., 54 (2005), 999.
doi: 10.1016/j.sysconle.2005.02.012. |
[9] |
Y. Chitour, M. Godoy Molina and P. Kokkonen, Symmetries of the rolling model, preprint,, \arXiv{1301.2579}., (). Google Scholar |
[10] |
Y. Chitour and P. Kokkonen, Rolling manifolds: Intrinsic formulation and controllability, preprint,, \arXiv{1011.2925}., (). Google Scholar |
[11] |
Y. Chitour and P. Kokkonen, Rolling manifolds on space forms,, Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire, 29 (2012), 927.
doi: 10.1016/j.anihpc.2012.05.005. |
[12] |
B. E. J. Dahlberg, The converse of the four vertex theorem,, Proc. Amer. Math. Soc., 133 (2005), 2131.
doi: 10.1090/S0002-9939-05-07788-9. |
[13] |
M. Godoy Molina, E. Grong, I. Markina and F. Silva Leite, An intrinsic formulation of the problem on rolling manifolds,, J. Dyn. Control Syst., 18 (2012), 181.
doi: 10.1007/s10883-012-9139-2. |
[14] |
E. Grong, Controllability of rolling without twisting or slipping in higher dimensions,, SIAM J. Control Optim., 50 (2012), 2462.
doi: 10.1137/110829581. |
[15] |
E. Hsu, "Stochastic Analysis on Manifolds,'' Graduate Studies in Mathematics 38,, American Mathematical Society, (2002).
|
[16] |
K. Hüper and F. Silva Leite, On the geometry of rolling and interpolation curves on $S^n$, $SO_n$ and Grassmann manifolds,, J. Dyn. Control Syst., 13 (2007), 467.
doi: 10.1007/s10883-007-9027-3. |
[17] |
B. D. Johnson, The nonholonomy of the rolling sphere,, Amer. Math. Monthly, 114 (2007), 500.
|
[18] |
V. Jurdjevic and J. A. Zimmerman, Rolling sphere problems on spaces of constant curvature,, Math. Proc. Cambridge Philos. Soc., 144 (2008), 729.
doi: 10.1017/S0305004108001084. |
[19] |
M. Levi, Geometric phases in the motion of rigid bodies,, Arch. Rational Mech. Anal., 122 (1993), 213.
doi: 10.1007/BF00380255. |
[20] |
K. Nomizu, Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact,, T\^ohoku Math. J., 30 (1978), 623.
doi: 10.2748/tmj/1178229921. |
[21] |
J. W. Robbin and D. A. Salamon, "Introduction to Differential Geometry,", Available from: \url{http://www.math.ethz.ch/\, (). Google Scholar |
[22] |
R. W. Sharpe, "Differential Geometry,'', GTM 166, (1997).
|
[23] |
M. Spivak, "A Comprehensive Introduction to Differential Geometry'',, Volume IV, (1999).
|
[24] |
I. Zelenko, On variational approach to differential invariants of rank two distributions,, Differential Geom. Appl., 24 (2006), 235.
doi: 10.1016/j.difgeo.2005.09.004. |
[25] |
I. Zelenko, Fundamental form and the Cartan tensor of $(2,5)$-distributions coincide,, J. Dyn. Control Syst., 12 (2006), 247.
doi: 10.1007/s10450-006-0383-1. |
[26] |
J. A. Zimmerman, Optimal control of the sphere $S^n$ rolling on $E^n$,, Math. Control Signals Systems, 17 (2005), 14.
doi: 10.1007/s00498-004-0143-2. |
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