zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Methods for determination and approximation of the domain of attraction. (English) Zbl 1066.34053
Consider the initial value problem $\frac{dx}{dt} =f(x) , x(0)=x^0$, where $f\in C^1(\Bbb R^n,\Bbb R^n)$ with $f(0)=0$. Assume that $x=0$ is asymptotically stable and $D_{a}(0)$ denotes the domain of attraction of 0. The authors introduce several approaches for the determination of $D_{a}(0)$. In order to determine and approximate $D_{a}(0)$ for an $R$-analytical system, an $R$-analytical function and the sequence of its Taylor polynomials are constructed by recurrence formula using the coefficients of the power series expansion of $f$ at 0. This construction differs from that one of {\it A. Vannelli} and {\it M. Vidyasagar} [Automatica 21, 69--80 (1985; Zbl 0559.34052)].

MSC:
34D20Stability of ODE
WorldCat.org
Full Text: DOI arXiv
References:
[1] Balint, S.: Considerations concerning the maneuvering of some physical systems. An. univ. Timisoara, seria st. Mat. 23, 8-16 (1985) · Zbl 0664.93039
[2] Balint, S.; Balint, A.; Negru, V.: The optimal Lyapunov function in diagonalizable case. An. univ. Timisoara, seria st. Mat. 24, 1-7 (1986) · Zbl 0615.34050
[3] Balint, S.; Negru, V.; Balint, A.; Simiantu, T.: An approach of the region of attraction by the region of convergence of the series of the optimal Lyapunov function. An. univ. Timisoara, seria st. Mat. 25, 15-30 (1987) · Zbl 0652.34065
[4] E. Barbashin, The method of sections in the theory of dynamical systems, Mat. Sb. 29. · Zbl 0131.31502
[5] Barbashin, E.; Krasovskii, N.: On the existence of Lyapunov functions in the case of asymptotic stability in the whole. Prikle. kat. Mekh. 18, 345-350 (1954)
[6] Gruyitch, L.; Richard, J. -P.; Borne, P.; Gentina, J. -C.: Stability domains, nonlinear systems in aviation, aerospace, aeronautics, astronautics. (2004)
[7] Kaslik, E.; Balint, A.; Balint, S.: Gradual approximation of the domain of attraction by gradual extension of the ”embryo” of the transformed optimal Lyapunov function. Nonlinear stud. 10, No. 1, 8-16 (2003) · Zbl 1042.34084
[8] Knobloch, H.; Kappel, F.: Gewohnliche differentialgleichungen. (1974) · Zbl 0283.34001
[9] Vanelli, A.; Vidyasagar, M.: Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. Automatica 21, No. 1, 69-80 (1985) · Zbl 0559.34052
[10] Zubov, V.: Methods of A.M. Lyapunov and their applications. (1964) · Zbl 0115.30204
[11] Zubov, V.: Théorie de la commande. (1978) · Zbl 0377.93002