Controllability of nilpotent systems

Victor Bravo

Banach Center Publications (1995)

  • Volume: 32, Issue: 1, page 35-46
  • ISSN: 0137-6934

Abstract

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In this paper we study the controllability property of invariant control systems on Lie groups. In [1], the authors state: ``If there exists a real function strictly increasing on the positive trajectories, then the system cannot be controllable". To develop this idea, the authors define the concept of symplectic vector via the co-adjoint representation. We are interested in finding algebraic conditions to determine the existence of symplectic vectors in nilpotent Lie algebras. In particular, we state a necessary and sufficient condition for controllability in the simply connected nilpotent case.

How to cite

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Bravo, Victor. "Controllability of nilpotent systems." Banach Center Publications 32.1 (1995): 35-46. <http://eudml.org/doc/262685>.

@article{Bravo1995,
abstract = {In this paper we study the controllability property of invariant control systems on Lie groups. In [1], the authors state: ``If there exists a real function strictly increasing on the positive trajectories, then the system cannot be controllable". To develop this idea, the authors define the concept of symplectic vector via the co-adjoint representation. We are interested in finding algebraic conditions to determine the existence of symplectic vectors in nilpotent Lie algebras. In particular, we state a necessary and sufficient condition for controllability in the simply connected nilpotent case.},
author = {Bravo, Victor},
journal = {Banach Center Publications},
keywords = {left-invariant vector fields; controllability; Lie algebra; nilpotent Lie group},
language = {eng},
number = {1},
pages = {35-46},
title = {Controllability of nilpotent systems},
url = {http://eudml.org/doc/262685},
volume = {32},
year = {1995},
}

TY - JOUR
AU - Bravo, Victor
TI - Controllability of nilpotent systems
JO - Banach Center Publications
PY - 1995
VL - 32
IS - 1
SP - 35
EP - 46
AB - In this paper we study the controllability property of invariant control systems on Lie groups. In [1], the authors state: ``If there exists a real function strictly increasing on the positive trajectories, then the system cannot be controllable". To develop this idea, the authors define the concept of symplectic vector via the co-adjoint representation. We are interested in finding algebraic conditions to determine the existence of symplectic vectors in nilpotent Lie algebras. In particular, we state a necessary and sufficient condition for controllability in the simply connected nilpotent case.
LA - eng
KW - left-invariant vector fields; controllability; Lie algebra; nilpotent Lie group
UR - http://eudml.org/doc/262685
ER -

References

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  1. [1] V. Ayala and L. Vergara, Co-adjoint representation and controllability, Proyecciones 11 (1992), 37-48. Zbl1082.93504
  2. [2] B. Bonnard, V. Jurdjevic, I. Kupka and G. Sallet, Transitivity of invariant vector fields on the semidirect product of Lie groups, Trans. Amer. Math. Soc. 271 (1982), 521-535. Zbl0519.49023
  3. [3] R. Brockett, Systems theory on group manifolds and coset spaces, SIAM J. Control 10 (1972), 265-284. Zbl0238.93001
  4. [4] L. Corwin and F. Greenleaf, Representations of Nilpotent Lie Groups and Their Applications, Part I, Cambridge University Press, 1990. Zbl0704.22007
  5. [5] V. Jurdjevic and I. Kupka, Control systems on semi-simple Lie groups and their homogeneous spaces, Ann. Inst. Fourier (Grenoble) 31 (4) (1981), 151-179. Zbl0453.93011
  6. [6] V. Jurdjevic and H. Sussmann, Control systems on Lie groups, J. Differential Equations 12 (1972), 313-329. Zbl0237.93027
  7. [7] I. Kupka, Introduction to the Theory of Systems, 16 Coloquio Brasileiro de Matematica, 1987. Zbl0606.49016
  8. [8] L. San Martin and P. Crouch, Controllability on principal fibre bundle with compact structure group, Systems Control Letters 5 (1984), 35-40. Zbl0562.93012
  9. [9] H. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188. 

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