### Abstract

We study the linear system {multiplication sign with dot above}=Ax+Bu from a differential geometric point of view. It is well-known that controllability of the system is related to the one-parameter family of operators e^{Λt}B. We use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in ℝ^{n}. We then view γ(t)=Im(e^{Λt}B) as a curve in appropriate Grassmannian and see that, in local coordinates, γ is an integral curve of the flow induced by a matrix Riccati equation. We obtain qualitative geometric conditions on γ that are equivalent to the controllability of the system. To get quantitiative results, we lift γ to a curve l' in a splitting space, a generalized Grassmannian, which has the advantage of being a reductive homogeneous space of the general linear group, GL(ℝ^{n}). Explicit and simple expressions concerning the geometry of Γ are computed in terms of the Lie algebra of GL(ℝ^{n}), and these are related to the controllability of the system.

Original language | English |
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Pages (from-to) | 281-317 |

Number of pages | 37 |

Journal | Acta Applicandae Mathematicae |

Volume | 16 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1989 |

### Keywords

- AMS subject classifications (1980): 93B27, 53C30
- Control
- Grassman manifold
- Riccati equation
- homogeneous space
- linear system

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## Cite this

*Acta Applicandae Mathematicae*,

*16*(3), 281-317. https://doi.org/10.1007/BF00052521