zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Systems on universe spaces. (English) Zbl 0826.93001
The concept of a control system is extended over a new type of state space called a universe space. The idea of the analytic universes is the following. The real line is completed with a phantom element $\emptyset_0$. Assigning $\emptyset_0$ to a partially defined function at each point outside its domain (for instance, $1/x = \emptyset_0$ if $x = 0$) one redefines the function globally. Denote $A_n$ the family of functions $F$ partially defined on $R^n$, analytic on their domains and extended in the sense described above. Then the analytic universe space $(X,C)$ is a pair of a set $X$ and a family $C$ of functions defined on $X$ and taking values in $R \cup \emptyset_0$, which is, in particular, close under analytic substitutions: if $c \in C^n$ and $F \in A_n$, then $F(c) \in C$. The family $C$ itself is called analytic function universe. Vector fields over universe spaces are introduced and local uniqueness of the trajectories is proved. Control systems are defined in terms of vector fields on universe spaces and the problem of observability is studied. Due to the definitions, the system itself and the observation functions may be only partially defined. It is shown that the indistinguishability of states of systems on universe spaces is transitive while classical indistinguishability is not. It is obtained that indistinguishable states are also infinitesimally indistinguishable; under appropriate assumptions the converse statement is proved as well. A useful tool for construction of an observable system from an unobservable one is passing to the quotient system. It is proved that quotient systems exist. For a given unobservable system an observable one with the same set of response maps is constructed.
Reviewer: D.Silin (Graz)

93A10General systems
12K99Generalizations of fields
Full Text: DOI
[1] Bartosiewicz, Z.: Observable systems from unobservable ones, in A. Isidori (ed.),Nonlinear Control System Design, IFAC Symp., Capri, Italy, 14-16 June 1989.
[2] Bartosiewicz, Z.: Rational systems and observation fields,Systems Control Lett. 9 (1987). · Zbl 0636.93040
[3] Bartosiewicz, Z.: Minimal polynomial realizations,Math. Control Signals Systems 1 (1988). · Zbl 0671.93005
[4] Bartosiewicz, Z.: Ordinary differential equations on real affine varieties,Bull. Polish Acad. Sci., Ser. Math. 15 (1987). · Zbl 0627.34007
[5] Diop, S. and Fliess, M.: On nonlinear observability, in C. Commaultet al. (eds),Proc. First European Conference, Hermès, Paris, 1991.
[6] Hermann, R. and Krener, A.: Nonlinear controllability and observability,IEEE Trans. AC-22 (1978). · Zbl 0396.93015
[7] Isidori, A.:Nonlinear Control Systems, Springer-Verlag, Berlin, 1985. · Zbl 0569.93034
[8] Johnson, J.: A generalized global differential calculus I,Cahiers Top. Geom. Diff. XXVII (1986), 25-83. · Zbl 0603.18004
[9] Sontag, E.:Polynomial Response Maps, Springer-Verlag, New York, 1979. · Zbl 0413.93004
[10] Sussmann, H.: Existence and uniqueness of minimal realizations of nonlinear systems,Math. Systems Theory 10 (1977).