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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)

##### MSC:
 93A10 General systems 93B07 Observability 12K99 Generalizations of fields
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##### References:
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