×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hermes, H.; LaSalle, J., Functional Analysis and Time-Optimal Control (1969), Academic Press: Academic Press New York · Zbl 0203.47504
[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) · Zbl 0242.49040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.