Short bio: The author of more than 250 technical papers and books, Chris Byrnes received an Honorary Doctorate of Technology from the Royal Institute of Technology (KTH) in Stockholm in 1998 and in 2002 was named a Foreign Member of the Royal Swedish Academy of Engineering Sciences. He is a Fellow of the IEEE and in 2005 was awarded the Reid Prize from SIAM for his contributions to Control Theory and Differential Equations. He will hold the Giovanni Prodi Chair in Nonlinear Analysis at the University of Wuerzburg in the summer of 2009 and spend the 2009-2010 academic year as Gast Professor at KTH, supported by the Swedish Strategic Research Foundation.

Abstract:A long term goal in the theory of systems and control is to develop a systematic
methodology for the design of feedback control schemes capable of shaping the re-
sponse of complex dynamical systems, in both an equilibrium and a nonequilibrium
setting. In this talk, we will focus primarily on periodic steady-state behavior, a phe-
nomenon that is pervasive in nature and in man-made systems. We will begin with an
analysis of Brockett's recent design of a feedback law which creates an asymptotically
stable oscillation in a three dimensional, nonholonomic model of an AC controlled
rotor with a constant steady-state angular velocity. We will show how to design
feedback laws for stabilizable n-dimensional systems so that the existence, periods
and stability of periodic responses can be analyzed and shaped when the nonlinear
feedback system is driven with an arbitrary periodic input.

This design is the result of joint work with joint work with R. Brockett and with A.
Isidori. The sufficient conditions use a multi-valued analogue of Liapunov functions,
in much the same way as the angular variable θ in polar coordinates is multi-valued.
For the AC motor the angular variable is the rotational component of the magnetic
field while for the two body problem with a central force field the existence of an
angular variable is a consequence of conservation of angular momentum. In the
general case, the sufficient conditions can be checked point-wise, just as in Liapunov
theory, without knowledge of the trajectories of the system or a cross-section for
the dynamics. Moreover, these techniques can be used to shape the Lagrange stable
orbits of dissipative nonlinear feedback systems, a key to solving problems of output
regulation. Finally, using the recent solution of the Poincar ́ Conjecture and more, we e show these sufficient conditions are necessary for the existence of an asymptotically
stable oscillation - similar in spirit of the converse theorems of Liapunov theory.