Dr. Joao Hespanha (UCSB Electrical and Computer Engineering)

Title: Stochastic Hybrid Systems: Applications and Mathematical Tools

Abstract: Hybrid dynamical systems combine continuous-time dynamics with discrete modes of operation. The states of such system usually have two distinct components: one that evolves continuously, typically according to a differential equation; and another one that only changes through instantaneous jumps. This talk addresses Stochastic Hybrid Systems (SHSs) where transitions between discrete modes are triggered by stochastic events, much like transitions between states of continuous-time Markov chains. However, in SHSs the rate at which transitions occur depends on both the continuous and the discrete states of the hybrid system.

The combination of continuous dynamics, discrete events, and stochasticity results in a modeling framework with tremendous expressive power, making SHSs appropriate to describe the dynamics of a wide variety of systems that arise in practical applications. This observation has been the driving force behind the search for mathematical tools that permit the formal analysis of SHSs. In this talk I will discuss some of the tools that have been most useful in analyzing and designing SHSs. Among these, we discuss the use of the extended generator, Lyapunov-based arguments, infinite-dimensional moment dynamics, and finite-dimensional truncations by moment closure.

We will illustrate the use of the analysis tools in the context of several applications of SHSs. We discuss examples that arise in distributed estimation and control as well as in population dynamics in molecular biology and ecology.