Viability theory
A Smooth Converse Lyapunov Theorem for Robust Stability
SIAM Journal on Control and Optimization
The Dynamics of Control
SIAM Journal on Control and Optimization
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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.