It is OK for dynamics to be multivalued and have nonunique solutions: from the leaky bucket ODE to the Conley decomposition and a total Lyapunov function for hybrid dynamics.

February 02, 2024, ESB 2001

Rafal Goebel

Loyola Univ. Chicago, Math


For natural generalizations of classical dynamics, like differential inclusions or hybrid dynamics, or due to accounting for perturbations or measurement error in a closed-loop system, continuous dependence of solutions on initial conditions may be too much to ask for. Concepts of set convergence and (semi)continuity of set-valued mappings come to the rescue. The talk briefly presents the foundations of these concepts and highlights how, under natural (semi)continuity assumptions on the data of a differential inclusion or a hybrid system, classical results on invariance, smooth Lyapunov functions, uniformity and robustness of asymptotic stability, etc., extend to these settings. The main part of the talk describes the Conley decomposition, sometimes called the fundamental theorem of dynamical systems, and shows how it extends from classical dynamics to (multivalued) hybrid dynamics. The decomposition accounts for all compact attractors inside a given invariant set and is established through the existence of a so-called total Lyapunov function.

Speaker's Bio

Rafal Goebel received his Ph.D. in mathematics in 2000 from the University of Washington. He held postdoctoral positions at the Departments of Mathematics at University of British Columbia and Simon Fraser University in Vancouver, and at the Electrical and Computer Engineering Department of University of California, Santa Barbara. In 2008, he joined the Department of Mathematics and Statistics at Loyola University Chicago. He received the 2009 SIAM Control and Systems Theory Prize and is a co-author of the Hybrid Dynamical Systems: Modeling, Stability, and Robustness book. His interests include convex, nonsmooth, and set-valued analysis; control, including optimal control; hybrid dynamical systems; mountains; and optimization.

Video URL: