Decision & Control: Convex analysis approach to the consensus problem

Abstract: The consensus/rendezvous problem is about controlling the agents in a multiagent system so that, asymptotically, all agents arrive at the same location, i.e., reach consensus. The challenges may lie in the agents communicating their current location only to their neighbors, in the communication structure switching over time, in constraints on the location of each agent, etc. Convex analysis, a modern but well-developed branch of mathematical analysis, deals with convex sets, including those with corners on the boundary, and convex functions, including nondifferentiable and infinite-valued ones. The introductory talk will show how elements of convex analysis can be used to establish consensus in a multiagent network, in the case of a symmetric communication graph, and subject to switching and to state constraints. This leads to a unification and simpler proofs of some existing results. A key idea is that an appropriate switching between gradient flows for several convex functions leads to convergence to a common minimizer of these functions if any such minimizers exist.

The talk is based on joint work with Ricardo Sanfelice.

Bio: 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 the 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 and at the Electrical and Computer Engineering Department at the University of California, Santa Barbara, 2002 -- 2005. In 2008 he joined the Department of Mathematics and Statistics at Loyola University Chicago, where he is currently a professor. His interests include convex, nonsmooth, and set-valued analysis; control theory, including optimal control; hybrid dynamical systems; mountains; and optimization.