Invariant Control Systems on Lie Groups: A Short Survey
Keywords:
Invariant control affine systems, detached feedback equivalence, optimal controlAbstract
This is a short survey of our recent research on invariant control systems (and their associated optimal control problems). We are primarily concerned with equivalence and classification, especially in three dimensions.
Downloads
References
R.M. Adams, R. Biggs, C.C. Remsing, Equivalence of control systems on the Euclidean group SE(2), Control Cybernet. 41 (3) (2012), 513 – 524.
R.M. Adams, R. Biggs, C.C. Remsing, On the equivalence of control systems on the orthogonal group SO (4), in “Recent Researches in Automatic Control, Systems Science and Communications” (H. R. Karimi, ed.), WSEAS Press, Porto, 2012, 54 – 59.
R.M. Adams, R. Biggs, C.C. Remsing, Single-input control systems on the Euclidean group SE (2), Eur. J. Pure Appl. Math. 5 (1) (2012), 1 – 15.
R.M. Adams, R. Biggs, C.C. Remsing, Control systems on the orthogonal group SO (4), Commun. Math. 21 (2) (2013), 107 – 128.
R.M. Adams, R. Biggs, C.C. Remsing, On some quadratic Hamilton-Poisson systems, Appl. Sci. 15 (2013), 1 – 12.
R.M. Adams, R. Biggs, C.C. Remsing, Two-input control systems on the Euclidean group SE (2), ESAIM Control Optim. Calc. Var. 19 (4) (2013), 947 – 975.
R.M. Adams, R. Biggs, C.C. Remsing, Quadratic Hamilton-Poisson systems on so∗−(3): classification and integration, in “Proccedings of the 15th International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2013” (I. M. Mladenov, A. Ludu, A. Yoshioka, eds.), Bulgarian Academy of Sciences, 2014, 55 – 66.
A.A. Agrachev, D. Barilari, Sub-Riemannian structures on 3D Lie groups, J. Dyn. Control Syst. 18 (1) (2012), 21 – 44.
A.A. Agrachev, Y.L. Sachkov, “Control Theory from the Geometric Viewpoint”, Encyclopaedia of Mathematical Sciences 87, Control Theory and Optimization II, Springer-Verlag, Berlin, 2004.
A. Aron, C. Dăniasă, M. Puta, Quadratic and homogeneous Hamilton-Poisson system on (so(3)), Int. J. Geom. Methods Mod. Phys. 4 (7) (2007), 1173 – 1186.
D.I. Barrett, R. Biggs, C.C. Remsing, Affine subspaces of the Lie algebra se (1, 1), Eur. J. Pure Appl. Math. 7 (2) (2014), 140 – 155.
D.I. Barrett, R. Biggs, C.C. Remsing, Optimal control of drift-free invariant control systems on the group of motions of the Minkowski plane, in “Proceedings of the 13th European Control Conference, Strasbourg, France, 2014”, European Control Association, 2014, 2466 – 2471.
D.I. Barrett, R. Biggs, C.C. Remsing, Quadratic Hamilton-Poisson systems on se (1, 1)∗− : the homogeneous case, Int. J. Geom. Methods Mod. Phys. 12 (1) (2015), 1550011.
C.E. Bartlett, R. Biggs, C.C. Remsing, Control systems on the Heisenberg group: equivalence and classification, Publ. Math. Debrecen 88 (1-2) (2016), 217 – 234.
J. Biggs, W. Holderbaum, Planning rigid body motions using elastic curves, Math. Control Signals Systems 20 (4) (2008), 351 – 367.
J. Biggs, W. Holderbaum, Integrable quadratic Hamiltonians on the Euclidean group of motions, J. Dyn. Control Syst. 16 (3) (2010), 301 – 317.
R. Biggs, P.T. Nagy, A classification of sub-Riemannian structures on the Heisenberg groups, Acta Polytech. Hungar. 10 (7) (2013), 41 – 52.
R. Biggs, C.C. Remsing, A category of control systems, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 20 (1) (2012), 355 – 367.
R. Biggs, C.C. Remsing, A note on the affine subspaces of three-dimensional Lie algebras, Bul. Acad. Ştiinţe Repub. Mold. Mat. 3 (70) (2012), 45 – 52.
R. Biggs, C.C. Remsing, On the equivalence of cost-extended control sys- tems on Lie groups, in ”Recent Researches in Automatic Control, Systems Science and Communications, Porto, Portugal, 2012” (H. R. Karimi, ed.), WSEAS Press, 2012, 60 – 65.
R. Biggs, C.C. Remsing, Control affine systems on semisimple three-dimensional Lie groups, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 59 (2) (2013), 399 – 414.
R. Biggs, C.C. Remsing, Control affine systems on solvable three-dimensional Lie groups, I, Arch. Math. (Brno) 49 (3) (2013), 187 – 197.
R. Biggs, C.C. Remsing, Control affine systems on solvable three-dimensional Lie groups, II, Note Mat. 33 (2) (2013), 19 – 31.
R. Biggs, C.C. Remsing, Feedback classification of invariant control systems on three-dimensional Lie groups, in “Proceedings of the 9th IFAC Symposium on Nonlinear Control Systems, Toulouse, France”, IFAC Proceedings Volumes 46 (23) (2013), 506 – 511.
R. Biggs, C.C. Remsing, A classification of quadratic Hamilton-Poisson systems in three dimensions, in “Proceedings of the 15th International Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, 2013” (I. M. Mladenov, A. Ludu, A. Yoshioka, eds.), Bulgarian Academy of Sciences, 2014, 67 – 78.
R. Biggs, C.C. Remsing, Control systems on three-dimensional Lie groups, in “Proceedings of the 13th European Control Conference, Strasbourg, France, 2014”, European Control Association, 2014, 2442 – 2447.
R. Biggs, C.C. Remsing, Control systems on three-dimensional Lie groups: equivalence and controllability, J. Dyn. Control Syst. 20 (3) (2014), 307 – 339.
R. Biggs, C.C. Remsing, Cost-extended control systems on Lie groups, Mediterr. J. Math. 11 (1) (2014), 193 – 215.
R. Biggs, C.C. Remsing, On the equivalence of control systems on Lie groups, Commun. Math. 23 (2) (2015), 119 – 129.
R. Biggs, C.C. Remsing, Equivalence of control systems on the pseudo- orthogonal group SO (2, 1), An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 24 (2) (2016), 45 – 65.
R. Biggs, C.C. Remsing, Quadratic Hamilton–Poisson systems in three dimensions: equivalence, stability, and integration, Acta Appl. Math. 148 (2017), 1 – 59.
R. Biggs, C.C. Remsing, Invariant Control Systems on Lie Groups, in “Lie Groups, Differential Equations, and Geometry: Advances and Surveys” (G. Falcone, ed.), Springer, in press.
A.M. Bloch, “Nonholonomic Mechanics and Control”, Springer-Verlag, New York, 2003.
A.M. Bloch, P.E. Crouch, N. Nordkvist, A.M. Sanyal, Embedded geodesic problems and optimal control for matrix Lie groups, J. Geom. Mech. 3 (2) (2011), 197 – 223.
R.W. Brockett, System theory on group manifolds and coset spaces, SIAM J. Control 10 (2) (1972), 265 – 284.
L. Capogna, E. Le Donne, Smoothness of sub-Riemannian isometries, Amer. J. Math. 138 (5) (2016), 1439 – 1454.
D. D’Alessandro, M. Dahleh, Optimal control of two-level quantum systems, IEEE Trans. Automat. Control 46 (6) (2001), 866 – 876.
C. Dăniasă, A. Gı̂rban, R.M. Tudoran, New aspects on the geometry and dynamics of quadratic Hamiltonian systems on (so(3))∗ , Int. J. Geom. Methods Mod. Phys. 8 (8) (2011), 1695 – 1721.
V.I. Elkin, Affine control systems: their equivalence, classification, quotient systems, and subsystems, J. Math. Sci. 88 (5) (1998), 675 – 721.
R.V. Gamkrelidze, Discovery of the maximum principle, J. Dyn. Control Syst. 5 (4) (1999), 437 – 451.
R.B. Gardner, W.F. Shadwick, Feedback equivalence of control systems, Systems Control Lett. 8 (5) (1987), 463 – 465.
V.V. Gorbatsevich, A.L. Onishchik, E.B. Vinberg, “Lie groups and Lie algebras III”, Springer-Verlag, Berlin, 1994.
S. Goyal, N.C. Perkins, C.L. Lee, Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables, J. Comput. Phys. 209 (1) (2005), 371 – 389.
K.Y. Ha, J.B. Lee, Left invariant metrics and curvatures on simply connected three-dimensional Lie groups, Math. Nachr. 282 (6) (2009), 868 – 898.
U. Hamenstädt, Some regularity theorems for Carnot-Carathéodory metrics, J. Differential Geom. 32 (3) (1990), 819 – 850.
A. Harvey, Automorphisms of the Bianchi model Lie groups, J. Math. Phys. 20 (2) (1979), 251 – 253.
B. Jakubczyk, Equivalence and Invariants of Nonlinear Control Systems, in “Nonlinear Controllability and Optimal Control” (H.J. Sussmann, ed.), Marcel Dekker, New York, 1990, 177 – 218.
B. Jakubczyk, W. Respondek, On linearization of control systems, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (9-10) (1980), 517 – 522.
V. Jurdjevic, The geometry of the plate-ball problem, Arch. Rational Mech. Anal. 124 (4) (1993), 305 – 328.
V. Jurdjevic, “Geometric Control Theory”, Cambridge University Press, Cambridge, 1997.
V. Jurdjevic, Optimal control on Lie groups and integrable Hamiltonian systems, Regul. Chaotic Dyn. 16 (5) (2011), 514 – 535.
V. Jurdjevic, F. Monroy-Pérez, Variational problems on Lie groups and their homogeneous spaces: elastic curves, tops, and constrained geodesic problems, in “Contemporary Trends in Nonlinear Geometric Control Theory and its Applications (México City, 2000)”, World Sci. Publ., River Edge, NJ, 2002, 3 – 51.
V. Jurdjevic, H.J. Sussmann, Control systems on Lie groups, J. Differential Equations 12 (1972), 313 – 329.
V. Jurdjevic, J. Zimmerman, Rolling sphere problems on spaces of constant curvature, Math. Proc. Cambridge Philos. Soc. 144 (3) (2008), 729 – 747.
I. Kishimoto, Geodesics and isometries of Carnot groups, J. Math. Kyoto Univ. 43 (3) (2003), 509 – 522.
V. Kivioja, E. Le Donne, Isometries of nilpotent metric groups, J. Éc. Polytech. Math. 4 (2017), 473 – 482.
A. Krasiński, C.G. Behr, E. Schücking, F.B. Estabrook, H.D. Wahlquist, G.F.R. Ellis, R. Jantzen, W. Kundt, The Bianchi classification in the Schücking-Behr approach, Gen. Relativity Gravitation 35 (3) (2003), 475 – 489.
A.J. Krener, On the equivalence of control systems and linearization of non-linear systems, SIAM J. Control 11 (1973), 670 – 676.
P.S. Krishnaprasad, “Optimal Control and Poisson Reduction”, Inst. Systems Research, University of Maryland, 1993, T.R. 93-87.
J. Lauret, Modified H-type groups and symmetric-like Riemannian spaces, Differential Geom. Appl. 10 (2) (1999), 121 – 143.
N.E. Leonard, P.S. Krishnaprasad, Motion control of drift-free, left-invariant systems on Lie groups, IEEE Trans. Automat. Control 40 (1995), 1539 – 1554.
M.A.H. MacCallum, On the classification of the real four-dimensional Lie algebras, in “On Einstein’s Path (New York, 1996)”, Springer-Verlag, New York, 1999, 299 – 317.
F. Monroy-Pérez, A. Anzaldo-Meneses, Optimal control on the Heisenberg group, J. Dyn. Control Syst. 5 (4) (1999), 473 – 499.
G.M. Mubarakzyanov, On solvable Lie algebras, Izv. Vysš. Učehn. Zaved. Matematika 1 (32) (1963), 114 – 123.
P.T. Nagy, M. Puta, Drift Lee-free left invariant control systems on SL(2, R) with fewer controls than state variables, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 44(92) (1) (2001), 3 – 11.
M. Puta, Stability and control in spacecraft dynamics, J. Lie Theory 7 (2) (1997), 269 – 278.
M. Puta, Optimal control problems on matrix Lie groups, in “New Developments in Differential Geometry”(J. Szenthe, ed.), Kluwer, Dordrecht, 1999, 361 – 373.
W. Respondek, I.A. Tall, Feedback Equivalence of Nonlinear Control Systems: a Survey on Formal Approach, in “Chaos in Automatic Control”, Control Engineering (Taylor & Francis), CRC Press, Boca Raton, FL, 2006, 137 – 262.
Yu.L. Sachkov, Conjugate points in the Euler elastic problem, J. Dyn. Control Syst. 14 (3) (2008), 409 – 439.
Yu.L. Sachkov, Maxwell strata in the Euler elastic problem, J. Dyn. Control Syst. 14 (2) (2008), 169 – 234.
Yu.L. Sachkov, Control theory on Lie groups, J. Math. Sci. 156 (3) (2009), 381 – 439.
H.J. Sussmann, An extension of a theorem of Nagano on transitive Lie algebras, Proc. Amer. Math. Soc. 45 (3) (1974), 349 – 356.
H.J. Sussmann, Lie Brackets, Real Analyticity and Geometric Control, in “Differential Geometric Control Theory” (R. W. Brockett, R. S. Millman, H. J. Sussmann, eds.), Birkhäuser, Boston, MA, 1983, 1 – 116.
R.M. Tudoran, The free rigid body dynamics: generalized versus classic, J. Math. Phys. 54 (7) (2013), 072704.
G.C. Walsh, R. Montgomery, S.S. Sastry, Optimal path planning on matrix Lie groups, in “Proc. 33rd Conf. Dec. & Control”, Lake Buena Vista 1994, 1258 – 1263.
E.N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (83) (1982), 337 – 346.
