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)
Sufficient conditions for viability under imperfect measurement. (English) Zbl 0802.49029

Given the output feedback system with uncertain measurement:

x ˙f(x(t),V(y(t)),x n ,yg(x(t),K),y m ,k m ,

one says that the mapping V(·): m U r is a viable regulation map with respect to the viability constraint x(t)X n , t[0,+) if for every x(0)X, x(t)X, t[0,+). One proves that if there is ε>0 and a constant ρ such that for every yY ε

min uU max xE ε (y) min z𝒫 X (x) f(x,u)-ρ(x-z),x-z0

the viable regulation map is given by:

V(y)={uU;max xE ε (y) [min z𝒫 X (x) f(x,u),x-z-ρdist(x,X) 2 ]0},yY ε ,U,yY ε ,

where Y ε =g(X+ε,K), E ε (y)={xX+ε; yg(x,K)}, with yY ε , and 𝒫 X (x)={x ' X; |x-x ' |=dist(x,X)}, with xX, and is the unit ball in n . Two examples illustrate the obtained results.

Reviewer: M.Voicu (Iaşi)
49N35Optimal feedback synthesis
93B52Feedback control
[1]Aubin, J.-P.:Viability Theory Birkhäuser, Boston, 1991.
[2]Aubin, J.-P. and Frankowska, H.:Set-Valued Analysis Birkhäuser, Boston, 1990.
[3]Clarke, F.:Optimization and Nonsmooth Analysis, Wiley, New York, 1983.
[4]Guseinov, K., Subotin, A., and Ushakov, V., Derivatives for multivalued mappings with applications to game-theoretical problems of control,Problems Control Inform. Theory 14 (1985), 155-167.
[5]Krasovskii, N.N. and Subbotin, A.I.:Positional Differential Games Nauka, Moscow, 1974; English transl.,Game-Theoretical Control Problems, Springer-Verlag, New York, 1988.
[6]Kurzhanski, A. and Nikonov, O., On adaptive processes in problems of guaranteed control,Izv. Akad. Nauk SSSR Techn. Kibernet. (4), 1986 (in Russian).
[7]Kurzhanski, A.B. and Nikonov, O.I.: On adaptive processes in problems of guaranteed control, inProc. 9th World Congress of IFAC, Vol. 5, Budapest, 1984, pp. 176-180.
[8]Veliov, V.: Guaranteed control of uncertain systems: Funnel equations and existence of regulation maps, Working Paper WP-92-62, International Institute for Applied Systems Analysis, Laxenburg, Austria, 1992.