Abstract
The theory of subanalytic sets is an excellent tool in various analytic-geometric contexts, including geometric control theory. (See [1], for example.)
One can axiomatize the notion of “behaving like the category of subanalytic sets (in manifolds)” by introducing the notion of “analyticgeometric category”. (The category of subanalytic sets is the smallest analytic-geometric category.) The objects of such a category share many of the hereditary and geometric finiteness properties of subanalytic sets. Proofs of the more difficult results of this nature, like the Whitney-stratifiability of sets and maps in such a category, often involve the use of charts to reduce to the case of subsets of ℝn. For subsets of ℝn, the theory of o-minimal structures on the real field, an abstraction of the theory of semialgebraic sets, provides an elegant and efficient setting in which to work. (See [2] and [3].)
(Some reasonable sets—like {(x, x r): x > 0} for irrational r, {(x, ex): x>0}, and {(x, Γ(x)): x > 0}—are not globally subanalytic in ℝ2. Because there are o-minimal structures on the real fieald which include these sets “at infinity”among their objects)
In analogy with the semilinear, semialgebraic and subanalytic settings, one considers hybrid systems whose relevant data (guards, resets, flows and so on) all belong to some o-minimal structure. It can be shown, for example, that such hybrid systems admit finite bisimulations; see [4].
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Differential Geometric Control Theory, Progr. Math. 27, Birkhäuser, Boston, 1983.
L. van den Dries, Tame Topology and O-minimal Structures, London Math. Soc. Lecture Note Series 248, Cambridge University Press, 1998.
L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84:497–540, 1996.
G. Lafferriere, G. Pappas, and S. Sastry, O-minimal hybrid systems, Technical Report, UC Berkeley, 1998. Available at http://www.mth.pdx.edu/~gerardo.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Miller, C. (1999). Geometric Categories, O-Minimal Structures and Control. In: Vaandrager, F.W., van Schuppen, J.H. (eds) Hybrid Systems: Computation and Control. HSCC 1999. Lecture Notes in Computer Science, vol 1569. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48983-5_4
Download citation
DOI: https://doi.org/10.1007/3-540-48983-5_4
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-65734-7
Online ISBN: 978-3-540-48983-2
eBook Packages: Springer Book Archive