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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)].

34D20Stability of ODE
Full Text: DOI arXiv
[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