Any two-input left-invariant control affine system of full rank, evolving on the Euclidean group SE (2), is (detached) feedback equivalent to one of three typical cases. In each case, we consider an optimal control problem which is then lifted, via the Pontryagin Maximum Principle, to a Hamiltonian system on the dual space 𝔰𝔢 (2)*. These reduced Hamilton - Poisson systems are the main topic of this paper. A qualitative analysis of each reduced system is performed. This analysis includes a study of the stability nature of all equilibrium states, as well as qualitative descriptions of all integral curves. Finally, the reduced Hamilton equations are explicitly integrated by Jacobi elliptic functions. Parametrisations for all integral curves are exhibited.
Keywords: left-invariant control system, (detached) feedback equivalence, Lie − Poisson structure, energy-casimir method, Jacobi elliptic function
@article{COCV_2013__19_4_947_0, author = {Adams, Ross M. and Biggs, Rory and Remsing, Claudiu C.}, title = {Two-input control systems on the euclidean group {~SE~(2)}}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {947--975}, publisher = {EDP-Sciences}, volume = {19}, number = {4}, year = {2013}, doi = {10.1051/cocv/2012040}, mrnumber = {3182676}, zbl = {1283.49003}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2012040/} }
TY - JOUR AU - Adams, Ross M. AU - Biggs, Rory AU - Remsing, Claudiu C. TI - Two-input control systems on the euclidean group SE (2) JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 947 EP - 975 VL - 19 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2012040/ DO - 10.1051/cocv/2012040 LA - en ID - COCV_2013__19_4_947_0 ER -
%0 Journal Article %A Adams, Ross M. %A Biggs, Rory %A Remsing, Claudiu C. %T Two-input control systems on the euclidean group SE (2) %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 947-975 %V 19 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2012040/ %R 10.1051/cocv/2012040 %G en %F COCV_2013__19_4_947_0
Adams, Ross M.; Biggs, Rory; Remsing, Claudiu C. Two-input control systems on the euclidean group SE (2). ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 4, pp. 947-975. doi : 10.1051/cocv/2012040. http://archive.numdam.org/articles/10.1051/cocv/2012040/
[1] Foundations of Mechanics, 2nd edition. Addison-Wesley (1978). | MR | Zbl
and ,[2] Single-input control systems on the Euclidean group SE (2). Eur. J. Pure Appl. Math. 5 (2012) 1-15. | MR
, and ,[3] Control Theory from the Geometric Viewpoint. Springer-Verlag (2004). | MR | Zbl
and ,[4] Elliptic Functions. Cambridge University Press (2006). | MR | Zbl
and ,[5] A category of control systems. An. Şt. Univ. Ovidius Constanţa 20 (2012) 355-368. | MR | Zbl
and ,[6] On the equivalence of control systems on Lie groups. Publ. Math. Debrecen (submitted).
and ,[7] On the equivalence of cost-extended control systems on Lie groups. Proc. 8th WSEAS Int. Conf. Dyn. Syst. Control. Porto, Portugal (2012) 60-65.
and ,[8] System theory on group manifolds and coset spaces. SIAM J. Control 10 (1972) 265-284. | MR | Zbl
,[9] Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1985) 1-116. | MR | Zbl
, , and ,[10] Non-Euclidean elastica. Amer. J. Math. 117 (1995) 93-124. | MR | Zbl
,[11] Geometric Control Theory. Cambridge University Press (1997). | MR | Zbl
,[12] Optimal control problems on Lie groups: crossroads between geometry and mechanics, in Geometry of Feedback and Optimal Control, edited by B. Jakubczyk and W. Respondek, M. Dekker (1998) 257-303. | MR | Zbl
,[13] Control systems on Lie groups. J. Differ. Equ. 12 (1972) 313-329. | MR | Zbl
and ,[14] Optimal control and Poisson reduction, Technical Research Report T.R.93-87. Inst. Systems Research, University of Maryland (1993).
,[15] Elliptic Functions and Applications. Springer-Verlag (1989). | MR | Zbl
,[16] Introduction to Mechanics and Symmetry. 2nd edition. Springer-Verlag (1999). | MR | Zbl
and ,[17] Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380-399. | Numdam | MR | Zbl
and ,[18] Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry. J. Geom. Phys. 32 (1999) 160-188. | MR | Zbl
and ,[19] Asymptotic and Lyapunov stability of constrained and Poisson equilibria. J. Differ. Equ. 214 (2005) 92-127. | MR | Zbl
, and ,[20] Differential Equations and Dynamical Systems, 3rd edition. Springer-Verlag (2001). | MR | Zbl
,[21] Hamiltonian Mechanical Systems and Geometric Quantization. Kluwer (1993). | MR | Zbl
,[22] An optimal control problem on the Lie group SE (2,R). Publ. Math. Debrecen 60 (2002) 15-22. | MR | Zbl
, and ,[23] Some remarks on an optimal control problem on the Lie group SE (2,R). An. Şt. Univ. A.I. Cuza Iaşi, ser. Mat. 49 (2003) 249-256. | MR | Zbl
, and ,[24] Optimal control and Hamilton − Poisson formalism. Int. J. Pure Appl. Math. 59 (2010) 11-17. | MR | Zbl
,[25] Control and stability on the Euclidean group SE (2). Lect. Notes Eng. Comput. Sci. Proc. WCE 2011. London, UK, 225-230.
,[26] Maxwell strata in the Euler elastic problem. J. Dyn. Control Syst. 14 (2008) 169-234. | MR | Zbl
,[27] Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018-1039. | Numdam | MR | Zbl
,[28] Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 17 (2011) 293-321. | Numdam | MR | Zbl
,Cited by Sources: