It is OK for (Hybrid) Dynamics to be Multivalued and Have Nonunique Solutions

Rafal Goebel is a professor of mathematics at Loyola University Chicago. He 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. 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.

Keynote Abstract

For natural generalizations of classical dynamics, that may be irregular, multivalued, or hybrid, uniqueness and continuous dependence of solutions on initial conditions and perturbations may be too much to ask for. Concepts and tools from set-valued analysis, which is a relatively modern branch of mathematical analysis, come to the rescue. The talk briefly presents some of these concepts and describes how they facilitate a natural extension of the classical asymptotic stability theory to multivalued and hybrid dynamics, in the setting of hybrid inclusions. The talk then highlights how the same foundations apply to other topics, including stability of switching systems, the existence of hybrid optimal controls, regularity of the Poincare mapping for hybrid inclusions, stochastic approximation of hybrid inclusions, and the Conley decomposition --- sometimes called the fundamental theorem of dynamical systems --- for hybrid dynamics.