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Two-input control systems on the Euclidean group SE (2)

Published online by Cambridge University Press:  04 July 2013

Ross M. Adams ,
Rory Biggs  and
Claudiu C. Remsing
Ross M. Adams
Affiliation:
Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za
Rory Biggs
Affiliation:
Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za
Claudiu C. Remsing
Affiliation:
Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown, South Africa. dros@webmail.co.za; rorybiggs@gmail.com; c.c.remsing@ru.ac.za
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Abstract

Any two-input left-invariant control affine system of full rank, evolving on theEuclidean group SE (2), is (detached) feedback equivalent to one ofthree typical cases. In each case, we consider an optimal control problem which is thenlifted, via the Pontryagin Maximum Principle, to a Hamiltonian system onthe dual space 𝔰𝔢 (2)*. These reduced Hamilton − Poisson systems are the maintopic of this paper. A qualitative analysis of each reduced system is performed. Thisanalysis includes a study of the stability nature of all equilibrium states, as well asqualitative descriptions of all integral curves. Finally, the reduced Hamilton equationsare explicitly integrated by Jacobi elliptic functions. Parametrisations for all integralcurves are exhibited.

Keywords

Left-invariant control system(detached) feedback equivalenceLie − Poisson structureenergy-Casimir methodJacobi elliptic function
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 947 - 975
Copyright
© EDP Sciences, SMAI, 2013

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References

R. Abraham and J.E. Marsden, Foundations of Mechanics, 2nd edition. Addison-Wesley (1978).
Adams, R.M., Biggs, R. and Remsing, C.C., Single-input control systems on the Euclidean group SE (2). Eur. J. Pure Appl. Math. 5 (2012) 1–15. Google Scholar
A.A. Agrachev and Y.L. Sachkov, Control Theory from the Geometric Viewpoint. Springer-Verlag (2004).
J.V. Armitage and W.F. Eberlein, Elliptic Functions. Cambridge University Press (2006).
Biggs, R. and Remsing, C.C., A category of control systems. An. Şt. Univ. Ovidius Constanţa 20 (2012) 355–368. Google Scholar
R. Biggs and C.C. Remsing, On the equivalence of control systems on Lie groups. Publ. Math. Debrecen (submitted).
R. Biggs and C.C. Remsing, On the equivalence of cost-extended control systems on Lie groups. Proc. 8th WSEAS Int. Conf. Dyn. Syst. Control. Porto, Portugal (2012) 60–65.
Brockett, R.W., System theory on group manifolds and coset spaces. SIAM J. Control 10 (1972) 265–284. Google Scholar
Holm, D.D., Marsden, J.E., Ratiu, T. and Weinstein, A., Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123 (1985) 1–116. Google Scholar
Jurdjevic, V., Non-Euclidean elastica. Amer. J. Math. 117 (1995) 93–124. Google Scholar
V. Jurdjevic, Geometric Control Theory. Cambridge University Press (1997).
V. Jurdjevic, 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.
Jurdjevic, V. and Sussmann, H.J., Control systems on Lie groups. J. Differ. Equ. 12 (1972) 313–329. Google Scholar
P.S. Krishnaprasad, Optimal control and Poisson reduction, Technical Research Report T.R.93–87. Inst. Systems Research, University of Maryland (1993).
D.F. Lawden, Elliptic Functions and Applications. Springer-Verlag (1989).
J.E. Marsden and T.S. Ratiu, Introduction to Mechanics and Symmetry. 2nd edition. Springer-Verlag (1999).
Moiseev, I. and Sachkov, Y.L., Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380–399. Google Scholar
Ortega, J-P. and Ratiu, T.S., Non-linear stability of singular relative periodic orbits in Hamiltonian systems with symmetry. J. Geom. Phys. 32 (1999) 160–188. Google Scholar
Ortega, J-P., Planas-Bielsa, V. and Ratiu, T.S., Asymptotic and Lyapunov stability of constrained and Poisson equilibria. J. Differ. Equ. 214 (2005) 92–127. Google Scholar
L. Perko, Differential Equations and Dynamical Systems, 3rd edition. Springer-Verlag (2001).
M. Puta, Hamiltonian Mechanical Systems and Geometric Quantization. Kluwer (1993).
Puta, M., Chirici, S. and Voitecovici, A., An optimal control problem on the Lie group SE (2,R). Publ. Math. Debrecen 60 (2002) 15–22. Google Scholar
Puta, M., Schwab, G. and Voitecovici, A., 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. Google Scholar
Remsing, C.C., Optimal control and Hamilton − Poisson formalism. Int. J. Pure Appl. Math. 59 (2010) 11–17. Google Scholar
C.C. Remsing, Control and stability on the Euclidean group    SE (2). Lect. Notes Eng. Comput. Sci. Proc. WCE 2011. London, UK, 225–230.
Sachkov, Y.L., Maxwell strata in the Euler elastic problem. J. Dyn. Control Syst. 14 (2008) 169–234. Google Scholar
Sachkov, Y.L., Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018–1039. Google Scholar
Sachkov, Y.L., Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 17 (2011) 293–321. Google Scholar