We consider control affine systems, as well as cost-extended control systems, on the (four-dimensional) Engel group. Specifically, we classify the full-rank left-invariant control affine systems (under both detached feedback equivalence and strongly detached feedback equivalence). The cost-extended control systems with quadratic cost are then classified (under cost equivalence), as are their associated Hamilton-Poisson systems (up to affine isomorphism). In all cases, we exhibit a complete list of equivalence class representatives.
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Adams RM, Biggs R, Remsing CC. Control systems on the orthogonal group
\(\mathsf {SO}(4)\). Commun Math 2013;21:107–28.
MathSciNetMATH
2.
Adams RM, Biggs R, Remsing CC. Equivalence of control systems on the Euclidean group
\(\mathsf {{{SE}}}(2)\). Control Cybernet 2012;41:513–24.
MathSciNetMATH
3.
Adams MA, Tie J. On sub-Riemannian geodesics on the Engel groups: Hamilton’s equations. Math Nachr 2013;286:1381–406.
MathSciNetMATH
4.
Agrachev AA, Barilari D. Sub-Riemannian structures on 3D Lie groups. J Dyn Control Syst 2012;18:21–44.
MathSciNetCrossRefMATH
5.
Agrachev AA, Sachkov YuL. Control Theory from the Geometric Viewpoint. Berlin: Springer; 2004.
CrossRefMATH
6.
Almeida DM. Sub-Riemannian homogeneous spaces of Engel type. J Dyn Control Syst 2014;20:149–66.
MathSciNetCrossRefMATH
7.
Ardentov A, Huang T, Sachkov YuL, Yang X. Extremals in the Engel group with a sub-Lorentzian metric. Preprint.
8.
Ardentov AA, Sachkov YuL. Conjugate points in nilpotent sub-Riemannian problem on the Engel group. J Math Sci 2013;195:369–90.
MathSciNetCrossRefMATH
9.
Ardentov AA, Sachkov YuL. Cut locus in the sub-Riemannian problem on Engel group. Dokl Math 2018;97:82–5.
CrossRefMATH
10.
Ardentov AA, Sachkov YuL. Cut time in sub-Riemannian problem on Engel group. ESAIM: COCV 2015;21:958–88.
MathSciNetCrossRefMATH
11.
Ardentov AA, Sachkov YuL. Extremal paths in the nilpotent sub-Riemannian problem on the Engel group (subcritical case of pendulum oscillations). J Math Sci 2014; 199:481–7.
MathSciNetCrossRefMATH
12.
Ardentov AA, Sachkov YuL. Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group. Mat Sb 2011;202:31–54.
MathSciNetCrossRefMATH
13.
Ardentov AA, Sachkov YuL. Maxwell strata and cut locus in sub-Riemannian problem on Engel group. Regul Chaot Dyn 2017;22:909–36.
MathSciNetCrossRef
14.
Barilari D, Boscain U, Sigalotti M (eds.) 2016. Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds. Eur. Math. Soc.
15.
Barrett DI, Biggs R, Remsing CC. Affine distributions on a four-dimensional extension of the semi-Euclidean group. Note Mat 2015;35:81–97.
MathSciNetMATH
16.
Barrett DI, Biggs R, Remsing CC. Affine subspaces of the Lie algebra
\(\mathfrak {se}(1,1)\). Eur J Pure Appl Math 2014;7:140–55.
MathSciNet
17.
Bartlett CE, Biggs R, Remsing CC. Control systems on nilpotent Lie groups of dimension
\(4\): equivalence and classification. Differential Geom Appl 2017;38:282–97.
MathSciNetCrossRefMATH
18.
Bartlett CE, Biggs R, Remsing CC. Control systems on the Heisenberg group: equivalence and classification. Publ Math Debrecens 2016;88:217–34.
MathSciNetCrossRefMATH
19.
Bellaïche A, Risler JJ, (eds). 1996. Sub-Riemannian Geometry. Basel: Birkhäuser.
MATH
20.
Beschastnyi I, Medvedev A. Left-invariant sub-Riemannian Engel structures: abnormal geodesics and integrability. Preprint.
21.
Biggs R, Nagy PT. A classification of sub-Riemannian structures on the Heisenberg groups. Acta Polytech Hungar 2013;10:41–52.
22.
Biggs R, Remsing CC. Control affine systems on semisimple three-dimensional Lie groups. An Şt Univ “A.I. Cuza” Iaşi Ser Mat. 2013;59:399–414.
MathSciNetMATH
23.
Biggs R, Remsing CC. Control affine systems on solvable three-dimensional Lie groups, I. Arch. Math. (Brno) 2013;43:187–97.
MathSciNetCrossRefMATH
24.
Biggs R, Remsing CC. Control affine systems on solvable three-dimensional Lie groups, II. Note Mat 2013;33:19–31.
MathSciNetMATH
25.
Biggs R, Remsing CC. Control systems on three-dimensional Lie groups: equivalence and controllability. J Dyn Control Syst 2014;20:307–39.
MathSciNetCrossRefMATH
26.
Biggs R, Remsing CC. Cost-extended control systems on Lie groups. Mediterr J Math 2014;11:193–215.
MathSciNetCrossRefMATH
27.
Biggs R, Remsing CC. Equivalence of control systems on the pseudo-orthogonal group
\(\mathsf {{{SO}}}(2,1)\). An Ştiinţ Univ Ovidius Constanţa 2016;24:45–65.
MATH
28.
Biggs R, Remsing CC. On the classification of real four-dimensional Lie groups. J Lie Theory 2016;26:1001–35.
MathSciNetMATH
29.
Biggs R, Remsing CC. On the equivalence of control systems on Lie groups. Commun Math 2015;23:119–29.
MathSciNetMATH
30.
Biggs R, Remsing CC. Some remarks on the oscillator group. Differential Geom Appl 2014;35:199–209.
MathSciNetCrossRefMATH
31.
Biggs R, Remsing CC. Subspaces of real four-dimensional Lie algebras: a classification of subalgebras, ideals, and full-rank subspaces. Extracta Math 2015;30:41–93.
MathSciNetMATH
32.
Cai Q, Huang T, Sachkov YuL, Yang X. Geodesics in the Engel group with sub-Lorentzian metric. J Dyn Control Syst 2016;22:465–483.
MathSciNetCrossRefMATH
33.
Calin O, Chang DC. Sub-Riemannian Geometry: General Theory and Examples. Cambridge: Cambridge University Press; 2009.
CrossRefMATH
34.
Jurdjevic V. Geometric Control Theory. Cambridge: Cambridge University Press; 1997.
MATH
35.
Knapp AW. Lie groups, Beyond an Introduction, 2nd ed. Boston: Birkhäuser; 2002.
MATH
36.
Laurent-Gengoux C, Pichereau A, Vanhaecke P. Poisson Structures. Heidelberg: Springer; 2013.
CrossRefMATH
37.
Moiseev I, Sachkov YuL. Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 2010;16:380–99.
MathSciNetCrossRefMATH
38.
Montgomery R. 2002. A Tour of Subriemannian Geometries, Their Geodesics and Applications. Amer. Math. Soc, Providence, RI.
39.
Sachkov YuL. Conjugate and cut time in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 2010;16:1018–39.
MathSciNetCrossRefMATH
40.
Sachkov YuL. Control theory on Lie groups. J Math Sci 2009;156:381–439.
MathSciNetCrossRefMATH
41.
Sachkov YuL. Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 2011;17:293–321.
MathSciNetCrossRefMATH
42.
Sachkov YuL. Symmetries of flat rank two distributions and sub-Riemannian structures. Trans Amer Math Soc 2004;356:457–94.
MathSciNetCrossRefMATH
43.
Stefani G, Boscain U, Gauthier J-P, Sarychev A, Sigalotti M, (eds). 2014. Geometric Control Theory and Sub-Riemannian Geometry. Heidelberg: Springer.
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