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Stabilization of controllable systems

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Sub-Riemannian Geometry

Part of the book series: Progress in Mathematics ((PM,volume 144))

Abstract

The goal of this paper is to present a partial survey on the local stabilizability of locally controllable systems.

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References

  1. A. A. Agrachev, Newton diagram and tangent cones to attainable sets, Colloque international sur l’Analyse des Systèmes contrôlés, Lyon, June 1990, Birkhäuser.

    Google Scholar 

  2. A. Andreini and A. Bacciotti, Stabilization by piecewise constant feedback, Proc. of the first European Control Conference, July 2–5 (1991), Vol. 1, p. 474–479.

    Google Scholar 

  3. A. A. Agrachev and R. V. Gamkrelidze, Local controllability and semigroups of diffeomorphisms, Acta Appl. Math., 32 (1993), no. 1, 1–57.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Aeyels, R. Sepulchre and I. Mareels, On the stabilization of planar homogeneous systems, Preprint, Université Catholique de Louvain, 1993.

    Google Scholar 

  5. A. Bacciotti, Local stabilization of nonlinear control systems, Series on advances in Mathematics for applied sciences, Vol. 8, World scientific, Singapore - New Jersey - London - Hong Kong, 1992.

    Google Scholar 

  6. A. Bacciotti, Linear feedback: the local and potentially global stabilization of cascade systems, Nolcos’92, M. Fliess ed., Bordeaux, June 92, p. 21–25.

    Google Scholar 

  7. C. I. Byrnes and A. Isidori, Asymptotic stabilization of minimum phase nonlinear systems, IEEE Trans. Aut Control, 36 (1991) p. 1122–1137.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992) p. 1024–1065.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. W. Brockett, Asymptotic stability and feedback stabilization, in: Differential geometric Control Theory, R. W. Brockett, R. S. Millman and H. J. Sussmann eds, Birkhäuser, Basel - Boston, 1983.

    Google Scholar 

  10. R. M. Bianchini and G. Stefani, Sufficient conditions of local controllability, Proc. of 25th Conference on Decision and Control, Athens, Greece, Dec. 1986, p. 967–970.

    Google Scholar 

  11. R. M. Bianchini and G. Stefani, Controllability along a trajectory: a variational approach, SIAM J. Control Optim., 31 (1993), no. 4, 900–927.

    Article  MathSciNet  MATH  Google Scholar 

  12. W. L. Chow, Über systeme von linearen partiellen differentialgleichungen ester Ordnung, Math. Ann., 117 (1940—41) p. 98–105.

    Article  Google Scholar 

  13. J.-M. Coron, A necessary condition for feedback stabilization, Systems and Control Letters, 14 (1990) p. 227–232.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.-M. Coron, Global asymptotic stabilization for controllable systems without drift, Math. Control Signals Systems, 5 (1992) p. 295–312.

    Article  MathSciNet  MATH  Google Scholar 

  15. J.-M. Coron, Links between local controllability and local continuous stabilization, Preprint Eth-Zürich and Université Paris-Sud, October 1991, and Nolcos’92, M. Fliess ed., Bordeaux, June 92, p. 477–482.

    Google Scholar 

  16. J.-M. Coron, Linearized control systems and applications to smooth stabilization, SIAM J. Control Optim., 32 (1994) p. 358–386.

    Article  MathSciNet  MATH  Google Scholar 

  17. J.-M. Coron, Relations entre commandabilité et stabilisations non linéaires, Collège de France Seminar, H. Brezis and J.-L. Lions eds., Research Notes in Math. Pitman, Boston, To appear.

    Google Scholar 

  18. J.-M. Coron, On the stabilization in finite time of locally controllable systems by means of continuous time-varying feedback laws, SIAM J. Control Optim. 33 (1995) p. 804–833.

    Article  MathSciNet  MATH  Google Scholar 

  19. J.-M. Coron, Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles bidimensionnels, C. R. Acad. Sci. Paris Ser.I Math., 317 (1993), no. 3, 271–276.

    MathSciNet  MATH  Google Scholar 

  20. J.-M. Coron and J.-B. Pomet, A remark on the design of time-varying stabilizing feedback laws for controllable systems without drift, Nolcos’92, M. Fliess ed., Bordeaux, June 1992, p. 413–417.

    Google Scholar 

  21. J.-M. Coron and L. Praly, Adding an integrator for the stabilization problem, Systems and Control Letters, 17 (1991) p. 89–104.

    Article  MathSciNet  MATH  Google Scholar 

  22. J.-M. Coron and L. Rosier, A relation between continuous time-varying and discontinuous feedback stabilization, J. Math. Syst., Estimation and Control (1993) To appear.

    Google Scholar 

  23. C. Canudas de Wit and O. J. Sordalen, Examples of piecewise smooth stabilization of driftless NL Systems with less inputs than states, Nolcos’92, Bordeaux, June 1990, M. Fliess ed., p. 26–30.

    Google Scholar 

  24. R. Chabour, G. Sallet, and J. C. Vivalda, Stabilization of nonlinear two dimensional systems, a bilinear approach, Nolcos’92, Bordeaux, June 1990, M. Fliess ed., p. 230–235.

    Google Scholar 

  25. W. P. Dayawansa, Recent advances in the stabilization problem for low dimensional systems, Nolcos’92, M. Fliess ed., Bordeaux, June 1992, p. 1–8.

    Google Scholar 

  26. W. P. Dayawansa and C. F. Martin, Asymptotic stabilization of law dimensional systems, in Nonlinear synthesis, Progress in Systems and Control Theory, Vol. 9, Birkhäuser, Boston, 1991.

    Google Scholar 

  27. W. P. Dayawansa, C. Martin, and G. Knowles, Asymptotic stabilization of a class of smooth two dimensional systems, SIAM J. Control Optim., 28 (1990) p. 1321–1349.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Fliess and F. Messager, Vers une stabilisation non linéaire discontinue, Proc. 9th Conf. Analysis and Optimization of Systems, Lecture Notes Control Inform., Springer, Berlin, 1990.

    Google Scholar 

  29. J. P. Gauthier, Structure des systèmes non-linéaires, Editions du CNRS, Paris, 1984.

    MATH  Google Scholar 

  30. M. Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, Bd. 9, Springer-Verlag, Berlin-Heidelberg, 1986.

    MATH  Google Scholar 

  31. A. Halanay, Solutions périodiques des systèmes non linéaires à petit paramètre, Atti délia Accademia Nazionale dei Lincei, 22 (1957), p. 30–32.

    MathSciNet  MATH  Google Scholar 

  32. R. S. Hamilton, The inverse function theorem of Nash and Moser, Bull Am. Math. Soc., 7 (1982) p. 65–222.

    Article  MathSciNet  MATH  Google Scholar 

  33. H. Hermes, Discontinuous vector fields and feedback control, in Differential equations and dynamical systems, J. K. Hale and J. P. La Salle eds., Academic Press, New York and London, 1967.

    Google Scholar 

  34. H. Hermes, On the synthesis of a stabilizing feedback control via Lie algebraic methods, SIAM J. Control Optim., 18 (1980) p. 352–361.

    Article  MathSciNet  MATH  Google Scholar 

  35. H. Hermes, Control systems which generate decomposable Lie algebras, J. Diff. Equations, 44 (1982) p. 166–187.

    Article  MathSciNet  MATH  Google Scholar 

  36. H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in Differential Equations, Stability and Control, S. Elaydi ed., Lecture Notes in Pure and Appl. Math., 127, Marcel Dekker, Inc., (1990) p. 249–260.

    Google Scholar 

  37. H. Hironaka, Subanalytic sets, in Number theory algebraic geometry and commutative algebra, in honor of Y. Akizuki, Kinokuniya Publications, Tokyo (1973) p. 453–493.

    Google Scholar 

  38. H. Hermes and M. Kawski, Local controllability of a single-input affine system, Nonlinear Anal, and Appl., Lecture Notes Pure and Appl. Math., 109 (1987) p. 235–248.

    MathSciNet  Google Scholar 

  39. A. Isidori, Nonlinear Control Systems, Springer-Verlag, Berlin Heidelberg, 1989.

    MATH  Google Scholar 

  40. A. Ilchmann, I. Nuernberger, and W. Schmale, Time-varying polynomial matrix systems, International J. Control, 40 (1984) p. 329–362.

    Article  MATH  Google Scholar 

  41. M. Kawski, Control variations with an increasing number of switchings, Bull. Amer. Math. Soc., 18 (1988) p. 149–152.

    Article  MathSciNet  MATH  Google Scholar 

  42. M. Kawski, Stabilization of nonlinear systems in the plane, Systems and Control Letters, 12 (1989) p. 169–175.

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Kawski, Higher-order small-time local controllability, in Nonlinear Controllability and Optimal Control, H. J. Sussmann ed., Monographs and Textbooks in Pure and Applied Mathematics, 113, Marcel Dekker, Inc., New York, 1990, p. 431–467.

    Google Scholar 

  44. M. Kawski, Homogeneous stabilizing feedback laws, Control Theory and Advanced Technology, 6 (1990) p. 497–516.

    MathSciNet  Google Scholar 

  45. A. V. Kazhikov, Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid, PMM U.S.S.R., 44 (1981) p. 672–674.

    Google Scholar 

  46. M. A. Krasnosel’Skii, The operator of translation along the trajectories of differential equations, Trans. Math. Monographs, 19, Providence, Amer. Math. Soc., 1968.

    Google Scholar 

  47. M. A. Krasnosel’Skii and P. P. Zabreiko, Geometrical Methods of nonlinear Analysis, Springer-Verlag, Berlin Heidelberg, 1984.

    Google Scholar 

  48. J.-L. Lions, Exact controllability, stabilization and perturbation for distributed systems, SIAM Rev., 30 (1988), p. 1–68.

    Article  MathSciNet  MATH  Google Scholar 

  49. J.-L. Lions, Contrôlabilité exacte, Masson, Paris, 1988.

    Google Scholar 

  50. J.-L. Lions, Are there connections between turbulence and controllability?, 9th IN-RIA International Conference, Antibes, June 12–15, 1990.

    Google Scholar 

  51. J.-L. Lions, Remarks on turbulence theory, Turbulence and Coherent Structures, O. Métais and M. Lesieur eds., Kluwer Academic Publishers, 1991, p. 1–13.

    Google Scholar 

  52. L. Landau and E. Lifchitz, Physique théorique, Mécanique, Ed. Mir, Moscou, 1988.

    Google Scholar 

  53. R.M. Murray, G. Walsh, and S.S. Sastry, Stabilization and tracking of nonholo-nomic control systems using time-varying state feedback, Nolcos 92, M. Fliess ed., Bordeaux, June 92, p. 182–187.

    Google Scholar 

  54. H. Nijmeier and A. J. Van Der Schaft, Nonlinear dynamical control systems, Springer-Verlag, New York, 1990.

    Google Scholar 

  55. R. Outbib and G. Sallet, Stabilizability of the angular velocity of a rigid body, Systems and Control Letters, 18 (1992) p. 93–98.

    Article  MathSciNet  MATH  Google Scholar 

  56. J.-B. Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems without drift, Systems and Control Letters, 18 (1992) p. 147–158.

    Article  MathSciNet  MATH  Google Scholar 

  57. L. Rosier, Homogeneous Lyapunov function for homogeneous vector field, Systems and Control Letters, 19 (1992) p. 467–473.

    Article  MathSciNet  MATH  Google Scholar 

  58. L. Rosier, Homogeneous stabilizable system which cannot be stabilizable by means of homogeneous feedback laws, Manuscript, December 1992.

    Google Scholar 

  59. D. L. Russell, Exact boundary value controllability theorems for wave and heat processes in star-complemented regions, in Differential Games and Control Theory, Roxin, Liu, and Sternberg eds., Marcel Dekker, New York, 1974, p. 291–319.

    Google Scholar 

  60. D. L. Russell, Controllability and stabilization theory for linear partial differential equations, Recent progress and open questions, SIAM Rev., 20 (1978) p. 639–679.

    Article  MathSciNet  MATH  Google Scholar 

  61. C. Samson, Velocity and torque feedback control of a nonholonomic art, Int. Workshop in Adaptative and Nonlinear Control: Issues in Robotics, Grenoble, 1990, p. 125–151, Lecture Notes in Control and Information Sciences, Vol. 162, Springer-Verlag, Berlin - New York.

    Google Scholar 

  62. R. Sepulchre, G. Campion, and V. Wertz, Some remarks about periodic feedback stabilization, Nolcos’92, M. Fliess ed., Bordeaux, June 92, p. 418–423.

    Google Scholar 

  63. H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Diff. Equations, 12 (1972) p. 95–116.

    Article  MathSciNet  MATH  Google Scholar 

  64. L. M. Silverman and H. E. Meadows, Controllability and observability in time variable linear systems, SIAM J. Control Optim., 5 (1967) p. 64–73.

    Article  MathSciNet  MATH  Google Scholar 

  65. E. D. Sontag, Finite dimensional open-loop control generators for nonlinear systems, Int. J. Control, 47 (1988) p. 537–556.

    Article  MathSciNet  MATH  Google Scholar 

  66. E. D. Sontag, A “universal” construction of Artstein’s theorem on nonlinear stabilization, Systems and Control Letters, 13 (1989) p. 117–123.

    Article  MathSciNet  MATH  Google Scholar 

  67. E. D. Sontag, Mathematical Control Theory, Deterministic finite dimensional Systems, Texts in Applied Mathematics 6, Springer-Verlag, New York - Berlin - Heidelberg - London - Paris - Tokyo - Hong Kong, 1990.

    Google Scholar 

  68. E. D. Sontag, Feedback stabilization of nonlinear systems, in Robust Control of linear Systems and nonlinear Control, M. A. Kaashoek, J. H. Van Schuppen and A. C. M. Ran eds., Birkhàuser, Cambridge, MA, 1990.

    Google Scholar 

  69. E. D. Sontag, Universal nonsingular controls, Systems and Control Letters, 19 (1992) p. 221–224.

    Article  MathSciNet  MATH  Google Scholar 

  70. E. D. Sontag and H. J. Sussmann, Remarks on continuous feedback, IEEE CDC, Albuquerque, 2 (1980) p. 916–921.

    Google Scholar 

  71. H. J. Sussmann, Subanalytic sets and feedback control, J. Diff. Equations, 31 (1979) p. 31–52.

    Article  MathSciNet  MATH  Google Scholar 

  72. H. J. Sussmann, Single-input observability of continuous-time systems, Math. Systems Theory, 12 (1979) p. 371–393.

    Article  MathSciNet  MATH  Google Scholar 

  73. H. J. Sussmann, Lie brackets and local controllability: a sufficient condition for scalar input systems, SIAM J. Control Optim., 21 (1983) p. 686–713.

    Article  MathSciNet  MATH  Google Scholar 

  74. H. J. Sussmann, A general theorem on local controllability, SIAM J. Control Optim., 25 (1987) p. 158–194.

    Article  MathSciNet  MATH  Google Scholar 

  75. A. Teel and L. Praly, Global stabilizability and observability implies semi-global stabilization by output feedbacks, Preprint, Fontainebleau, December 1992.

    Google Scholar 

  76. A. I. Tret’Yak, On odd-order necessary conditions for optimality in a time-optimal control problem for systems linear in the control, Math. U.S.S.R. Sbornik, 70 (1991) p. 47–63.

    Article  MathSciNet  MATH  Google Scholar 

  77. J. Tsinias, Sufficient Lyapunov-like conditions for stabilization, Math. Control Signals Systems, 2 (1989) p. 343–357.

    Article  MathSciNet  MATH  Google Scholar 

  78. J. Tsinias, Stabilization of nonlinear systems by using integrators, Nolcos’92, M. Fliess ed., Bordeaux, June 92, p. 645–648.

    Google Scholar 

  79. S. H. Wang, Stabilization of decentralized control systems via time-varying controllers, IEEE Trans. Aut. Control, AC 27 (1982) p. 741–744.

    Article  MATH  Google Scholar 

  80. G. W. Whitehead, Elements of homotopy theory, Graduate text in Mathematics, 61, Springer-Verlag, New York Inc., 1978.

    MATH  Google Scholar 

  81. J. Zabczyk, Some comments on stabilizability, Applied Mathematics and Optimization, 19 (1989) p. 1–9

    Article  MathSciNet  MATH  Google Scholar 

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Coron, JM. (1996). Stabilization of controllable systems. In: Bellaïche, A., Risler, JJ. (eds) Sub-Riemannian Geometry. Progress in Mathematics, vol 144. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9210-0_5

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  • DOI: https://doi.org/10.1007/978-3-0348-9210-0_5

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-0348-9946-8

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