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)
Stabilizability of linear systems over a commutative normed algebra with applications to spatially distributed and parameter-dependent systems. (English) Zbl 0564.93054
The authors study the feedback stabilization of linear systems given by a pair of matrices (F,G) with entries in a commutative normed ${\bbfK}$- algebra $B\sb 0$ with identity. The problem is to find a suitable feedback matrix L over $B\sb 0$ such that the closed loop system is stable. The system is transformed into another system $(\hat F,\hat G)$ over a commutative $C\sp*$-algebra, where ${\hat{\ }}$ means the Gelfand transform. Necessary and sufficient conditions for the stabilizability of $(\hat F,\hat G)$ are obtained in terms of the corresponding Riccati equation. If the image $\hat B\sb 0$ or $\hat B $(B: the completion of $B\sb 0)$ under the Gelfand transform is *-closed, it is shown that the stabilizability of (F,G) is equivalent to that of $(\hat F,\hat G)$. Another condition for the stabilizability of (F,G) is stated in terms of local stabilizability of the system which is equivalent to the local rank condition for $(\hat F,\hat G)$. An example is shown as to the positioning of a seismic cable which is written by a discrete-time linear equation.
Reviewer: T.Nambu

93D15Stabilization of systems by feedback
93B25Algebraic theory of control systems
93C05Linear control systems
44A15Special transforms (Legendre, Hilbert, etc.)
46H25Normed modules and Banach modules, topological modules
93B17System transformation
93C25Control systems in abstract spaces
Full Text: DOI