## UNLV Theses, Dissertations, Professional Papers, and Capstones

12-1-2014

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

Zhonghai Ding

David Costa

Hossein Tehrani

Pushkin Kachroo

107

#### Abstract

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 Lazer-McKenna 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^1-smooth, the nonlinearity in the Lazer-McKenna suspension bridge equation is not C^1-smooth, which makes the exact controllability problem 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.

#### Keywords

Actuators; Bridge failures – Prevention; Control; PDEs; Oscillations; Piezoelectric devices; Suspension bridges

#### Disciplines

Civil Engineering | Mathematics | Transportation Engineering

English

COinS