Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
left-invariant control system; detached feedback equivalence; orthogonal group
left-invariant control system; detached feedback equivalence; orthogonal group
Summary:
We classify the left-invariant control affine systems evolving on the orthogonal group $SO(4)$. The equivalence relation under consideration is detached feedback equivalence. Each possible number of inputs is considered; both the homogeneous and inhomogeneous systems are covered. A complete list of class representatives is identified and controllability of each representative system is determined.
References:
[1] Adams, R.M., Biggs, R., Remsing, C.C.: On the equivalence of control systems on the orthogonal group SO(4). Recent Researches in Automatic Control, Systems Science and Communications, 2012, 54-59, WSEAS Press. MR 2881500
[2] Adams, R.M., Biggs, R., Remsing, C.C.: Equivalence of control systems on the Euclidean group SE(2). Control Cybernet., 41, 2012, 513-524. MR 3087026
[3] Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometric Viewpoint. 2004, Springer. MR 2062547 | Zbl 1062.93001
[4] Aron, A., Moş, I., Csaky, A., Puta, M.: An optimal control problem on the Lie group SO(4). Int. J. Geom. Methods Mod. Phys., 5, 2008, 319-327. DOI 10.1142/S0219887808002795 | MR 2422030 | Zbl 1159.49002
[5] Biggs, J.D., Holderbaum, W.: Singularities of optimal control problems on some six dimensional Lie groups. IEEE Trans. Automat. Control, 52, 2007, 1027-1038. DOI 10.1109/TAC.2007.899010 | MR 2329893
[6] Biggs, R., Remsing, C.C.: A category of control systems. An. Şt. Univ. Ovidius Constanţa, 20, 2012, 355-368. MR 2928428 | Zbl 1274.93062
[7] Biggs, R., Remsing, C.C.: A note on the affine subspaces of three-dimensional Lie algebras. Bul. Acad. Ştiinţe Repub. Mold. Mat., 3, 2012, 45-52. MR 3155842
[8] Biggs, R., Remsing, C.C.: On the equivalence of cost-extended control systems on Lie groups. Recent Researches in Automatic Control, Systems Science and Communications, 2012, 60-65, WSEAS Press.
[9] Biggs, R., Remsing, C.C.: Control affine systems on semisimple three-dimensional Lie groups. An. Ştiinţe. Univ. Al. I. Cuza Iaşi. Mat., 59, 2013, 399-414.
[10] Biggs, R., Remsing, C.C.: Cost-extended control systems on Lie groups. Mediterr. J. Math., To appear in Mediterr. J. Math. DOI: 10.1007/s00009-013-0355-0.
[11] Biggs, R., Remsing, C.C.: Equivalence of control systems on the pseudo-orthogonal group $SO(2,1)_0$. preprint.
[12] Biggs, R., Remsing, C.C.: On the equivalence of control systems on Lie groups. preprint.
[13] Birtea, P., Caşu, I., Raţiu, T.S., Turhan, M.: Stability of equilibria for the $\mathfrak{so}(4)$ free rigid body. J. Nonlinear Sci., 22, 2012, 187-212. DOI 10.1007/s00332-011-9113-2 | MR 2912325
[14] Bogoyavlensky, O.I.: Integrable Euler equations on SO(4) and their physical applications. Commun. Math. Phys., 93, 1984, 417-436. DOI 10.1007/BF01258538 | MR 0745694
[15] D'Alessandro, D.: The optimal control problem on SO(4) and its applications to quantum control. IEEE Trans. Automat. Control, 47, 2002, 87-92. DOI 10.1109/9.981724 | MR 1879692
[16] Isidori, A.: Nonlinear Control Systems (Third Edition). 1995, Springer.
[17] Jakubczyk, B.: Equivalence and invariants of nonlinear control systems. Nonlinear Controllability and Optimal Control, 1990, 177-218, Marcel Dekker. MR 1061386 | Zbl 0712.93027
[18] Jovanović, B.: Non-holonomic geodesic flows on Lie groups and the integrable Suslov problem on SO(4). J. Phys. A: Math. Gen., 31, 1998, 1415-1422. DOI 10.1088/0305-4470/31/5/011 | MR 1628570 | Zbl 0945.70014
[19] Jurdjevic, V.: Geometric Control Theory. 1997, Cambridge University Press. MR 1425878 | Zbl 0940.93005
[20] Jurdjevic, V., Sussmann, H.J.: Control systems on Lie groups. J. Diff. Equations, 12, 1972, 313-329. DOI 10.1016/0022-0396(72)90035-6 | MR 0331185 | Zbl 0237.93027
[21] Knapp, A. W.: Lie Groups Beyond an Introduction (Second Edition). 2005, Birkhäuser. MR 1920389
[22] Komarov, I.V., Kuznetsov, V.B.: Kowalewski's top on the Lie algebras $\mathfrak{o} (4)$, $\mathfrak{e} (3)$ and $\mathfrak{o} (3,1)$. J. Phys. A: Math. Gen., 23, 1990, 841-846. DOI 10.1088/0305-4470/23/6/010 | MR 1048764
[23] Linton, C., Holderbaum, W., Biggs, J.: Rigid body trajectories in different 6D spaces. ISRN Math. Phys., 2012. DOI 10.5402/2012/467520
[24] Nijmeijer, H., Schaft, A. van der: Nonlinear Dynamical Control Systems. 1996, Springer. MR 1047663
[25] Respondek, W., Tall, I.A.: Feedback equivalence of nonlinear control systems: a survey on formal approach. Chaos in Automatic Control, 2006, 137-262, CRC Press. MR 2283271 | Zbl 1203.93039
[26] Sachkov, Y.L.: Control theory on Lie groups. J. Math. Sci., 156, 2009, 381-439. DOI 10.1007/s10958-008-9275-0 | MR 2373391 | Zbl 1211.93038
[27] Sokolov, V.V., Wolf, T.: Integrable quadratic classical Hamiltonians on $\mathfrak{so} (4)$ and $\mathfrak{so} (1,3)$. J. Phys. A: Math. Gen., 39, 2006, 1915-1926. DOI 10.1088/0305-4470/39/8/009 | MR 2209308
[28] Sussmann, H.J.: Lie brackets, real analyticity and geometric control. Differential Geometric Control Theory, 1983, 1-116, Birkhäuser. MR 0708500 | Zbl 0545.93002
Search
Partner of
