Open Collections

Pointwise Asymptotic Stability

Banff International Research Station for Mathematical Innovation and Discovery

The talk presents some concepts and results from systems and control theory, focusing on convergence to and stability of a continuum of equilibria in a dynamical system.
The well-studied and understood asymptotic stability of a compact set requires Lyapunov stability of the set: solutions that start close remain close to the set, and that every solution converge to the set, in terms of distance. Pointwise asymptotic stability of a set of equilibria requires Lyapunov stability of each equilibrium, and that every solution converge to one of the equilibria. This property is present, for example, in continuous-time steepest descent and convergent saddle-point dynamics, in optimization algorithms generating convergent Fejer monotone sequences, etc., and also in many consensus algorithms for multi-agent systems. The talk will present some background on asymptotic stability and then discuss necessary and sufficient conditions for pointwise asymptotic stability in terms of set-valued Lyapunov functions; robustness of this property to perturbations; and how the property can be achieved in a control system by optimal control.

Mathematics; Operations research, mathematical programming; Operator theory; Control/optimization/operation research

video/mp4

Author affiliation: Loyola University Chicago

2018-03-27

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64943

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/