BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Pointwise Asymptotic Stability Goebel, Rafal


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.

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