Smooth Morse-Lyapunov Functions of Strong Attractors for Differential Inclusions
Smooth Morse-Lyapunov Functions of Strong Attractors for Differential Inclusions
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DOI：
10.1137/10081280x
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发表时间：
2012
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影响因子：
--
通讯作者：
Desheng Li;Yanling Wang
中科院分区：
文献类型：
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作者：
Desheng Li;Yanling Wang
本文研究了差分包含$x'(t)\in F(x(t))$强吸引子的Morse分解的光滑逆Lyapunov定理，其中$F$是$\mathbb{R}^m$上具有紧凸值的上半连续多值映射。粗略地说，假设有一个具有吸引盆$\Omega$和莫尔斯分解$\mathcal{M}=\{M_1,\ldots,M_l\}$的系统的强吸引子$\mathscr{A}$。我们将构造一个径向无界函数$V\in C^\infty(\Omega)$，使得(1)$V$在每个莫尔斯集合$M_k$上是常数，(2)$V$沿着系统在$\Omega$的莫尔斯集合外的任何解严格递减。
This paper is concerned with a smooth converse Lyapunov theorem for Morse decompositions of strong attractors of differential inclusion $x'(t)\in F(x(t))$, where $F$ is an upper semicontinuous multivalued mapping on $\mathbb{R}^m$ with compact convex values. Roughly speaking, let there be given a strong attractor $\mathscr{A}$ of the system with attraction basin $\Omega$ and Morse decomposition $\mathcal{M}=\{M_1,\ldots,M_l\}$. We will construct a radially unbounded function $V\in C^\infty(\Omega)$ such that (1) $V$ is constant on each Morse set $M_k$ and (2) $V$ is strictly decreasing along any solution of the system in $\Omega$ outside the Morse sets.