Control systems subordinated to a group action: Accessibility.

*(English)*Zbl 0531.93008Let M be an n-dimensional real \(C^{\infty}\) manifold; a polysystem on M is a family of \(C^{\infty}\) vector fields on M. Let G be a real connected Lie group which acts smoothly on M and let \(\Gamma\) be a subset of the Lie algebra of G. \(\Gamma\) defines a polysystem on M, the polysystem of the vector fields canonically induced by the elements of \(\Gamma\). Such a polysystem is called subordinated to a group action. A polysystem is transitive (controllable) on M if for each x, y belonging to M, there exists a trajectory of the polysystem which steers x to y. The authors give sufficient conditions for transitivity of a polysystem on GL(V) (V a finite dimensional vector space) associated to \(\Gamma =\{A+uB:\) \(u\in R\), A,\(B\in End(V)\}\).

Reviewer: R.M.Bianchini

##### MSC:

93B05 | Controllability |

57R27 | Controllability of vector fields on \(C^\infty\) and real-analytic manifolds |

93B03 | Attainable sets, reachability |

57R25 | Vector fields, frame fields in differential topology |

22E60 | Lie algebras of Lie groups |

93C10 | Nonlinear systems in control theory |

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\textit{V. Jurdjevic} and \textit{I. Kupka}, J. Differ. Equations 39, 186--211 (1981; Zbl 0531.93008)

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##### References:

[1] | Hermes, H; LaSalle, J, Functional analysis and time-optimal control, (1969), Academic Press New York · Zbl 0203.47504 |

[2] | {\scV. Jurdjevic and I. Kupka}, Control systems: Accessibility on semi-simple Lie groups and their homogenous spaces, to appear. · Zbl 0453.93011 |

[3] | Jurdjevic, V; Quinn, J.P, Controllability and stability, J. differential equations, 28, 381-389, (1976) · Zbl 0417.93012 |

[4] | Sussman, H.J; Jurdjevic, V, Controllability of non-linear systems, J. differential equations, 12, 95-116, (1972) |

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