Decision, Control and Optimization Seminar

Decision and Control Laboratory, Coordinated Science Laboratory

Infinite Chains of Dynamical Systems

Professor Bruce Francis

University of Toronto, Dept. of ECE

Wednesday, April 4, 2012

3:00 PM to 4:00 PM

B02 CSL

Abstract
      An interesting application of autonomous mobile robotics is the convoying of vehicles: A lead vehicle is driven by a person and the others follow in line controlled autonomously, each pursuing the one ahead. In studying the formation of a very large number of vehicles, one approach is instead to model an infinite number of vehicles. This line of research goes back at least to Melzer and Kuo in 1971, includes Bamieh, Paganini and Dahleh in 2002, then D'Andrea and Dullerud in 2003, and up to Curtain, Iftime and Zwart in 2010. The relevant question is what mathematical framework to take so that the infinite-chain model correctly describes the behavior of the large-but-finite chain model.

Related systems arise in physics. In the Principia, 1687, Newton derived a formula for the speed of sound in an elastic medium. This was the first study of the so-called “loaded string” problem, now called the “one-dimensional lattice problem.” Newton's model had discrete masses representing the molecules -- pde's had not been invented yet. An infinite chain of masses are lined up in a string, connected by springs. Much more recently, in the book Wave Propagation in Periodic Media, 1948, Brillouin modeled a crystal by an infinite inductor/capacitor ladder. These two physics models are also examples of an infinite chain of dynamical systems.

In this talk we look at such infinite-chain systems. Such chains can be described mathematically by ordinary differential equations on Hilbert or Banach spaces, and questions of existence and stability of solutions are of interest. The answers are sometimes surprising.

Biography
Bruce Francis was born in Toronto and obtained his B.A.Sc. and M.Eng. degrees in Mechanical Engineering and his Ph.D. degree in Electrical Engineering from the University of Toronto in 1969, 1970, and 1975, respectively. He received a two-year postdoctoral fellowship in 1975. He spent the first year in the EECS Department at the University of California, Berkeley, and the second year in the Control and Management Systems Division, University of Cambridge. In 1977 he received postdoctoral support in Montreal from George Zames at McGill and M. Vidyasagar at Concordia. This turned into a faculty position in the EE Department at McGill during 1978-79. He spent 1979-81 as an assistant professor in the Engineering and Applied Science Department, Yale University, and returned to Canada in 1981 as an assistant professor in the EE Department, University of Waterloo. Finally, he returned to  the University of Toronto in 1984 and was promoted to full professor in 1986. He retired on December 31, 2011, and now holds the position of Professor Emeritus.
 

PLEASE JOIN US FOR COOKIES AND COFFEE PRIOR TO THE SEMINAR

IN ROOM 154 CSL AT 2:40PM