A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.
We study the linear system ,~ = 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 eafB. Wc use this to give a proof of the classical controllability conditions in terms of the differential geometry of certain curves in R ~. We then view ",/(t)= lm(e^'B) as a curve in appropriate Grassmannian and see that, in local coordinates, 3' is an integral curve of the tlow induced by a matrix Riccati equation. Wc obtain qualitative geometric conditions on 3" that are equiwdcnt to the controllability of the system. To get quantitiative results, we lift 3" to a curve I" in a splitting space, a generalized Grassmannian, which has the advantage of being a reduetive homogeneous space of the general linear group, GL(R"). Explicit and simple expressions concerning the geometry of F arc computed in terms of the Lie algebra of GL(R"), and these are related to the controllability of the system.
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