Speaker: Rafal Goebel (Loyola University at Chicago)
Title: It is OK for dynamics to be multivalued, have nonunique solutions, and blend continuous time with discrete time.
Abstract: For natural generalizations of classical dynamics, like differential inclusions, where ``the right-hand side is multivalued'', or hybrid dynamics, which blend continuous-time dynamics with discrete-time dynamics --- or due to accounting for perturbations or measurement error in an engineered system, the uniqueness and continuous dependence of solutions on initial conditions may be too much to ask for. Much of the asymptotic stability theory for these generalizations can be built with the help of set-valued analysis concepts and results. The talk surveys the concepts that make this happen, with the key property being the outer/upper-semicontinuous dependence of solutions on initial conditions for a differential inclusion or a hybrid inclusion. (The said semicontinuous dependence already occurs for $\dot{x}=\sqrt{x}$, the ``leaky bucket'' dynamics that show up in an undergraduate ODE class to illustrate the nonuniqueness of solutions.) The talk then discusses Morse and Conley decompositions of a compact attractor in a hybrid inclusion and highlights their relevance for stochastic approximation of a hybrid inclusion.