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Zbl 0559.34052
Vannelli, A.; Vidyasagar, M.
Maximal Lyapunov functions and domains of attraction for autonomous nonlinear systems. (English)
[J] Automatica 21, 69-80 (1985). ISSN 0005-1098

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).
[G.Ikegami]

MSC 2000:: *34D20 Lyapunov stability of ODE

Keywords: computational methods; domain of attraction; autonomous nonlinear system; asymptotically stable equilibrium point

Cited in: Zbl 1066.34053

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Zentralblatt MATH Berlin [Germany]