Consensus problems analyze systems where the dynamics of several autonomous agents cause the agents to converge, or not, to a common state. Equivalently, one may ask whether the system consisting of these several agents converges to a consensus state. Different applications motivate the analysis of dynamical systems which possess a continuum of equilibrium states. In such systems, each of the equilibrium states cannot be asymptotically stable and this leads to a concept of pointwise asymptotic stability (also called semistability) of the set of equilibria. It turns out that commonly applied sufficient conditions for convergence to a consensus in a multi-agent system, expressed in terms of “decreasing sets'', ensure pointwise asymptotic stability of the set of consensus states. These concepts, and the relationships between them, motivate this talk.
The talk will present Lyapunov-like necessary and sufficient conditions for pointwise asymptotic stability, expressed in terms of decreasing Lyapunov-like set-valued mappings, rather than in terms of classical Lyapunov functions. Efforts will be made to underline the similarities to the classical theory, as long as one accepts to use set inclusions in place of inequalities. An invariance principle, in terms of a non-increasing set-valued mapping, will also be given. The existence of a strictly decreasing Lyapunov-like set-valued mapping for a pointwise asymptotically stable set will be shown, in the spirit of classical converse Lyapunov theorems. Further connections between consensus, pointwise asymptotic stability, and the usual asymptotic stability will be discussed.

Rafal Goebel received his M.Sc. degree in mathematics in 1994 from the University of Maria Curie Sklodowska in Lublin, Poland, and his Ph.D. degree in mathematics in 2000 from University of Washington, Seattle. He held a postdoctoral position at the Departments of Mathematics at the University of British Columbia and Simon Fraser University in Vancouver, Canada, 2000 -- 2002; a postdoctoral and part-time research positions at the Electrical and Computer Engineering Department at the University of California, Santa Barbara, 2002 -- 2005; and a part-time teaching position at the Department of Mathematics at the University of Washington, 2004 -- 2007 In 2008, he joined the Department of Mathematics and Statistics at Loyola University Chicago. His interests include convex, nonsmooth, and set-valued analysis; control theory, including optimal control; hybrid dynamical systems; and optimization.