On a new differential geometric setting in nonlinear control (Q1365705)

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''.