Exact Controllability of the Lazer-McKenna Suspension Bridge Equation

EXACT CONTROLLABILITY OF THE LAZER-MCKENNA SUSPENSION BRIDGE EQUATION by Lanxuan Yu Dr. Zhonghai Ding, Examination Committee Chair Professor, Mathematics University of Nevada, Las Vegas, USA It is well known that suspension bridges may display certain oscillations under external aerodynamic forces. Since the collapse of the Tacoma Narrows suspension bridge in 1940, suspension bridge models have been studied by many researchers. Based upon the fundamental nonlinearity in suspension bridges that the stays connecting the supporting cables and the roadbed resist expansion, but do not resist compression, new models describing oscillations in suspension bridges have been developed by Lazer and McKenna [Lazer and McKenna (1990)]. Except for a paper by Leiva [Leiva (2005)], there have been very few work on controls of the LazerMcKenna suspension bridge models in the existing literature. In this dissertation, I use the Hilbert Uniqueness Method and the Leray-Schauder’s degree theory to study two exact controllability problems of the Lazer-McKenna suspension bridge equation. The first problem is to study the exact controllability of the single Lazer-McKenna suspension bridge equation with a locally distributed control. Unlike most of the existing literatures on exact controllability of nonlinear systems where the nonlinearity was always assumed to be C-smooth, the nonlinearity in the Lazer-McKenna suspension bridge equation is not C-smooth, which makes the exact controllability problem iii challenging to study. It is proved that the control system is exactly controllable. The key step is to establish an observability inequality of the auxiliary linear control problem. The proof of such an inequality relies on deriving a Carleman estimate. The second problem studied in this dissertation is the exact controllability problem of the single Lazer-McKenna suspension bridge equation with a piezoelectric bending actuator. It is proved that the control system is exactly controllable when the location of the actuator is carefully chosen. The proof of exact controllability is based upon establishing an Ingham inequality for nonharmonic Fourier series. iv ACKNOWLEDGEMENTS First, I want to give my most sincere thanks to my advisor, Dr. Zhonghai Ding for his guidance, patience, and most importantly, his inspiration during my graduate studies. His mentorship and encouragement was indispensable to the completion of this dissertation. At the same time, his attitudes towards research provided me with a living example, which I will aspire to follow in my future career. I would like to thank Dr. David Costa, Dr. Hossein Tehrani and Dr. Pushkin Kachroo for serving as my dissertation committee. Their advice and comments are so valuable. I am also very grateful to all the professors who have taught me for all the classes they taught and their mentoring for all these years at UNLV. v TABLE OF CONTENTS ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . v LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii CHAPTER