On a new differential geometric setting in nonlinear control
type: Item
instance of: scholarly article
title: On a new differential geometric setting in nonlinear control
zbMATH Open document ID: 0885.93014
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-09-08T00:00:00Z
review text: Apparently the authors deal with infinite-dimensional manifolds, because they would like to have a formalism which facilitates considering `all' -- i.e. a countable set of -- derivatives of the control variables, but they do it their own way. On the countable Cartesian product of \(\mathbb{R}\) they want to introduce a differentiable structure by means of defining the class of \(C^\infty\)-functions by saying that these ones ``depend only on a finite number of variables.'' But the inverse transformation of a \(C^\infty\)-coordinate transformation (in this sense) does not necessarily have the same property (Example: \(y_n= x_{n+1} -nx_n)\). Therefore this procedure does not lead to any geometry. Similarly, it cannot be seen that ``the definition of vector fields, differential forms ... is straightforward''.
Mathematics Subject Classification ID: 93B29
Mathematics Subject Classification ID: 93C10
zbMATH DE Number: 1058638
zbMATH Keywords: \(C^ \infty\)-coordinate transformation
zbMATH Keywords: infinite-dimensional manifolds
zbMATH Keywords: differentiable structure
author: Michel Fliess
author: Jean Lévine
author: Philippe Martin
author: Pierre Rouchon
reviewed by: Peter Kraut
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1058638