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Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. (English) Zbl 0559.34052
This paper presents theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system in ${\bbfR}\sp n$ (1) $\dot x($t)$=f(x(t))$ such that $x=0$ is an asymptotically stable equilibrium point. Define the domain of attraction by $S\equiv \{x\sb 0: x(t,x\sb 0)\to 0$ as $t\to \infty \}$, where $x(\cdot,x\sb 0)$ denotes the solution of (1) with $x(0)=x\sb 0$. The following is a theoretical result: Suppose f is Lipschitz continuous on the domain of attraction S for (1). Then, an open set A containing 0 coincides with S, if and only if there exist a continuous function $V: A\to {\bbfR}\sb+$ and a function with positive definite function $\phi$ such that: (i) $V(0)=0$, $V(x)>0$ for all $x\in A\setminus 0$. (ii) The function $\dot V(x\sb 0)=\lim\sb{t\to 0+}[V(x(t,x\sb 0))-V(x\sb 0)]t\sp{-1}$ is well defined at every $x\sb 0\in A$ and satisfies $\dot V($x)$=-\phi (x)$ for all $x\in A$. (iii) V(x)$\to \infty$ as $x\to \partial A$ and/or as $\vert x\vert \to \infty$. This implies that if we find a function v and a positive definite function $\phi$ satisfying $V(0)=0$ and $\dot V($x)$=-\phi (x)$ on some neighborhood of 0 then $\partial S$ is defined by V(x)$\to \infty$. The authors show a systematic procedure for solving $\dot V($x)$=-\phi (x)$ in the case of where f is an analytic function. It is shown that this equation is simpler than that of V. I. Zubov (1984).
Reviewer: G.Ikegami

MSC:
34D20Stability of ODE
WorldCat.org
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References:
[1] Best, M. J.: FCDPAK: A Fortran IV subroutine to solve differentiable mathematical programmes--user’s guide--level 3.1. Research report CORR 75-24 (1975)
[2] Best, M. J.; Bowler, A. T.: ACDPAC: A Fortran IV subroutine to solve differentiable mathematical programmes--user’s guide--level 2.0. Research report CORR 75-26 (1978)
[3] Brockett, R. W.: The status of stability theory for deterministic systems. IEEE trans. Aut. control 11, 596 (1966)
[4] Brown, K. M.; Jr., J. E. Dennis: Derivative free analogues of the Levenberg-Marquardt and Gauss algorithms for nonlinear least squares approximation. Numer. math. 18, 289 (1972) · Zbl 0235.65043
[5] Davison, E. J.; Kurak, E. M.: A computational method for determining quadratic Lyapunov functions for nonlinear systems. Automatica 7, 627 (1971) · Zbl 0225.34027
[6] Gatto, M.; Rinaldi, S.: Stability analysis of predator-prey models via the Lyapunov method. Analysis and computation of equilibria and regions of stability (1975)
[7] Hahn, W.: Stability of motion. (1967) · Zbl 0189.38503
[8] Langenhop, C. E.: Bounds on the norm of a solution of a general differential equation. Proc. amer. Math. soc. 11, 795 (1960) · Zbl 0102.08101
[9] Lasalle, J. P.; Lefschetz, S.: Stability of Lyapunov’s direct method with applications. (1961)
[10] Levenberg, K.: A method for the solution of certain nonlinear problems in the least squares. Quart. appl. Math. 2, 164 (1944) · Zbl 0063.03501
[11] Loparo, K. A.; Blankenship, G. L.: Estimating the domain of attraction of nonlinear feedback systems. IEEE trans. Aut. control 23, 602 (1978) · Zbl 0385.93023
[12] Malkin, I. G.: On the question of the reciprocal of Lyapunov’s theorem on asymptotic stability. Prinkl. mat. Meh. 18, 129 (1954)
[13] Marquardt, D. W.: An algorithm for least squares estimation of nonlinear parameters. SIAM J. Appl. math. 11, 431 (1963) · Zbl 0112.10505
[14] Massera, J. L.: On Lyapunov’s conditions of stability. Ann. math.. 50, 705 (1949) · Zbl 0038.25003
[15] Michel, A. N.; Miller, R. K.; Nam, B. H.: Stability analysis of interconnected systems using computer generated Lyapunov functions. IEEE trans. Circuits syst. 29, 431 (1982) · Zbl 0492.93049
[16] Pai, M. A.; Mohan, M. A.; Rao, J. Gopala: Power system transient stability regions using Popov’s method. IEEE trans. Power app. Syst. 89, 788 (1970)
[17] Schultz, D. G.: The generation of Lyapunov functions. Advances in control systems 2 (1965)
[18] Vidyasagar, M.: Nonlinear systems analysis. (1978) · Zbl 0407.93037
[19] Walker, J. A.; Clamroch, H. N.: Finite regions of attraction for the problem of Lur’e. Int. J. Control 6, 331 (1967)
[20] Weissengerger, S.: Application of results from the absolute stability problem to the computation of finite stability domains. IEEE trans. Aut. control 13, 124 (1968)
[21] Willems, J. L.: Improved Lyapunov functions for transient power-system stability. Proc. inst. Elec. eng. 115, 1315 (1968) · Zbl 0176.06901
[22] Willems, J. L.: Direct methods for transient stability studies in power system analysis. IEEE trans. Aut. control 16, 332 (1971)
[23] Yu, Y.; Vongsuriya, K.: Nonlinear power system stability study by Lyapunov function and zubov’s method. IEEE trans. Power appl. Syst. 86, 1480 (1967)
[24] Zubov, V. I.: Methods of A. M. Lyapunov and their application. (1964) · Zbl 0115.30204