Real analytic geometry and local observability.

*(English)* Zbl 0968.93015
Ferreyra, G. (ed.) et al., Differential geometry and control. Proceedings of the Summer Research Institute, Boulder, CO, USA, June 29-July 19, 1997. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 64, 65-72 (1999).

From the author’s abstract: “The language of real analytic geometry is used to give necessary and sufficient conditions of stable local observability of analytic systems… Globalization of this result yields a new characterization of local observability at large which uses the concept of sheaf.”
The notion of indistinguishability and theorem 3.1 giving necessary and sufficient conditions for this property are the basic material of the paper. Theorem 3.1 is taken from a paper of other authors [{\it R. Hermann} and {\it A. Krener}, Nonlinear controllability and observability, IEEE Trans. Autom. Control AC-22, 728-740 (1977;

Zbl 0396.93015)] without proof. The definition of an algebra ${\cal H}$ of real analytic functions by means of which the conditions above are expressed makes the reviewer have his doubts about the correctness of the theorem: the definition of indistinguishability also contains some conditions concerning the control parameter; conclusions resulting from the corresponding partial derivatives seem to have been forgotten.
In the fourth chapter (“Globalization”) the author “uses the concept of sheaf” in the following way. “H is identified with the set of sections of the sheaf $x\to{\cal H}_x$…” Obviously he has not become aware of the problem of completeness of the presheaf ${\cal H}$. For the entire collection see [

Zbl 0903.00046].

##### MSC:

93B07 | Observability |

93B27 | Geometric methods in systems theory |

14P15 | Real analytic and semianalytic sets |

93B29 | Differential-geometric methods in systems theory (MSC2000) |