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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

34D20Stability of ODE
Full Text: DOI
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