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Control systems subordinated to a group action: Accessibility. (English) Zbl 0531.93008
Let 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
##### Keywords:
bilinear systems; polysystem; transitivity
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##### References:
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