Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 095, 28 pages      arXiv:0903.4932      https://doi.org/10.3842/SIGMA.2009.095
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Geometry of Control-Affine Systems

Jeanne N. Clelland a, Christopher G. Moseley b and George R. Wilkens c
a) Department of Mathematics, 395 UCB, University of Colorado, Boulder, CO 80309-0395, USA
b) Department of Mathematics and Statistics, Calvin College, Grand Rapids, MI 49546, USA
c) Department of Mathematics, University of Hawaii at Manoa, 2565 McCarthy Mall, Honolulu, HI 96822-2273, USA

Received April 02, 2009, in final form September 28, 2009; Published online October 07, 2009

Abstract
Motivated by control-affine systems in optimal control theory, we introduce the notion of a point-affine distribution on a manifold X – i.e., an affine distribution F together with a distinguished vector field contained in F. We compute local invariants for point-affine distributions of constant type when dim(X) = n, rank(F) = n–1, and when dim(X) = 3, rank(F) = 1. Unlike linear distributions, which are characterized by integer-valued invariants – namely, the rank and growth vector – when dim(X) ≤ 4, we find local invariants depending on arbitrary functions even for rank 1 point-affine distributions on manifolds of dimension 2.

Key words: affine distributions; control theory; exterior differential systems; Cartan's method of equivalence.

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References

  1. Agrachev A.A., Feedback-invariant optimal control theory and differential geometry. II. Jacobi curves for singular extremals, J. Dynam. Control Systems 4 (1998), 583-604.
  2. Agrachev A.A., Sarychev A.V., Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 635-690.
  3. Agrachev A., Zelenko I., On feedback classification of control-affine systems with one- and two-dimensional inputs, SIAM J. Control Optim. 46 (2007), 1431-1460, math.OC/0502031.
  4. Bryant R.L., Chern S.S., Gardner R.B., Goldschmidt H.L., Griffiths P.A., Exterior differential systems, Mathematical Sciences Research Institute Publications, Vol. 18, Springer-Verlag, New York, 1991.
  5. Bryant R.L., Conformal geometry and 3-plane fields on 6-manifolds, in Developments of Cartan Geometry and Related Mathematical Problems, RIMS Symposium Proceedings, Vol. 1502, Kyoto University, 2006, 1-15, math.DG/0511110.
  6. Bullo F., Lewis A.D., Geometric control of mechanical systems. Modeling, analysis, and design for simple mechanical control systems, Texts in Applied Mathematics, Vol. 49, Springer-Verlag, New York, 2005.
  7. Cartan É., Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910), 109-192.
  8. Clelland J.N., Moseley C.G., Sub-Finsler geometry in dimension three, Differential Geom. Appl. 24 (2006), 628-651, math.DG/0406439.
  9. Clelland J.N., Moseley C.G., Wilkens G.R., Geometry of sub-Finsler Engel manifolds, Asian J. Math. 11 (2007), 699-726.
  10. Doubrov B., Zelenko I., On local geometry of nonholonomic rank 2 distributions, math.DG/0703662.
  11. Doubrov B., Zelenko I., On local geometry of rank 3 distributions with 6-dimensional square, arXiv:0807.3267.
  12. Elkin V.I., Affine control systems: their equivalence, classification, quotient systems, and subsystems. Optimization and control, 1, J. Math. Sci. (New York) 88 (1998), 675-721.
  13. Elkin V.I., Reduction of nonlinear control systems. A differential geometric approach, Mathematics and its Applications, Vol. 472, Kluwer Academic Publishers, Dordrecht, 1999.
  14. Ernst R., Bodenhausen G., Wokaun A., Principles of nuclear magnetic resonance in one and two dimensions, Oxford University Press, Oxford, 1987.
  15. Gardner R.B., The method of equivalence and its applications, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 58, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989.
  16. Ivey T.A., Landsberg J.M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics, Vol. 61, American Mathematical Society, Providence, RI, 2003.
  17. Jakubczyk B., Respondek W., Feedback classification of analytic control systems in the plane, in Analysis of Controlled Dynamical Systems (Lyon, 1990), Progr. Systems Control Theory, Vol. 8, Birkhäuser Boston, Boston, MA, 1991, 263-273.
  18. Kang W., Extended controller form and invariants of nonlinear control systems with a single input, J. Math. Systems Estim. Control 6 (1996), 27-51.
  19. Kang W., Krener A.J., Extended quadratic controller normal form and dynamic state feedback linearization of nonlinear systems, SIAM J. Control Optim. 30 (1992), 1319-1337.
  20. Liu W., Sussman H.J., Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564, 104 pages.
  21. Montgomery R., A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, Vol. 91, American Mathematical Society, Providence, RI, 2002.
  22. Nielsen M., Chuang I., Quantum computation and quantum information, Cambridge University Press, Cambridge, 2000.
  23. Respondek W., Feedback classification of nonlinear control systems on R2 and R3, in Geometry of Feedback and Optimal Control, Monogr. Textbooks Pure Appl. Math., Vol. 207, Dekker, New York, 1998, 347-381.
  24. Tall I.A., Respondek W., Feedback classification of nonlinear single-input control systems with controllable linearization: normal forms, canonical forms, and invariants, SIAM J. Control Optim. 41 (2002), 1498-1531.
  25. Wilkens G.R., Centro-affine geometry in the plane and feedback invariants of two-state scalar control systems, in Differential Geometry and Control (Boulder, CO, 1997), Proc. Sympos. Pure Math., Vol. 64, Amer. Math. Soc., Providence, RI, 1999, 319-333.
  26. Zhitomirskii M., Respondek W., Simple germs of corank one affine distributions, Singularities Symposium - Lojasiewicz 70 (Kraków, 1996; Warsaw, 1996), Banach Center Publ., Vol. 44, Polish Acad. Sci., Warsaw, 1998, 269-276.

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