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


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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