A Glimpse at Pointwise Asymptotic Stability for Continuous-Time and Discrete-Time Dynamics

  • Rafal GoebelEmail author


Given a dynamical system, pointwise asymptotic stability, also called semistability, of a set requires that every point in the set be a Lyapunov stable equilibrium, and that every solution converge to one of the equilibria in the set. This note provides examples of pointwise asymptotic stability related to optimization and states select results from the literature, focusing on necessary and sufficient Lyapunov and Lyapunov-like conditions for and robustness of this stability property. Background on the classical asymptotic stability is included.


Pointwise asymptotic stability Differential inclusion Difference inclusion Monotone operator Set-valued Lyapunov function 

AMS 2010 Subject Classification

93D05 49J53 90C25 34D20 47H05 



This work was partially supported by the Simons Foundation Grant 315326.


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Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLoyola University ChicagoChicagoUSA

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