Asymptotic Properties of Nonlinear Feedback Control Systems

This page is a substitute for the real page 2, the signature page. I am inserting it so that subsequent page numbers will appear on the correct side in the top margin. iii ACKNOWLEDGEMENTS First of all, I would like to express my deep gratitude to my advisor, Professor Roger W. Brockett of Harvard University. His boundless knowledge has made it possible for me to write this thesis, and his unfailing enthusiasm has made it impossible not to write it. I consider myself extremely lucky to have had the opportunity to work with him. I wish to acknowledge the help of two other people that has directly contributed to this dissertation. Mark Adler has always been willing to share his insights with me, and I am indebted to him for many illuminating discussions of various areas of mathematics related to this thesis. Susan Parker has provided helpful comments on my writing, and has communicated to me her excitement about teaching, which has been of great value in getting through my moments of frustration. My sincere thanks go to everyone at the Mathematics Department of Brandeis University for creating such a comfortable and stimulating atmosphere. I wish all of you the best of luck in surviving the diicult times. I dedicate this thesis to my family, who started me out on this journey and have always been supportive. We study asymptotic behaviour of nonlinear feedback control systems, both deterministic and stochastic. Of particular interest is the case of quantized feedback , i.e., when the nonlinearity takes the form of a speciic piecewise constant function. In the context of deterministic linear control systems with quantized measurements, we show how quantized feedback can be used to asymptotically stabilize the system (Chapter II). For systems perturbed by white noise, we address the question of existence of steady-state probability distributions. In the linear case, the solution to the v Fokker-Planck equation which describes the evolution of the probability density is well known. In particular, one has an expression for the steady-state probability density, which is an eigenfunction of the Fokker-Planck operator with eigenvalue zero. We show that other eigenvalues and eigenfunctions of the Fokker-Planck operator associated with a linear system can also be directly computed (Chapter III). In the nonlinear case, the situation is more complicated. We describe a class of nonlinear feedback systems for which explicit formulae for the steady-state probability densities can be …