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-Schauderu27s 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.
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机译：众所周知，在外部空气动力作用下，悬索桥可能会表现出一定的振动。自1940年塔科马海峡悬索桥倒塌以来，悬索桥模型已为许多研究人员研究。基于悬索桥的基本非线性，即保持支撑缆线和路基的抗扩展性，但不抵抗压缩，因此，Lazer和McKenna开发了描述悬索桥振动的新模型[Lazer and McKenna（1990）]。除了Leiva的论文[Leiva（2005）]，在现有文献中，关于Lazer-McKenna悬索桥模型控制的工作很少。本文采用希尔伯特唯一性方法和Leray-Schauder学位理论研究了Lazer-McKenna悬索桥方程的两个精确可控性问题。第一个问题是研究单个Lazer-McKenna悬架的精确可控性。具有局部分布控制的桥梁方程。与大多数关于非线性系统的精确可控性的现有文献不同，非线性总是被假定为C ^ 1-光滑，Lazer-McKenna悬索桥方程中的非线性不是C ^ 1-光滑，这导致了精确的可控性问题挑战学习。实践证明，该控制系统是完全可控的。关键步骤是建立辅助线性控制问题的可观测性不等式。这种不等式的证明依赖于对Carleman估计的推导。本文研究的第二个问题是带有压电弯曲致动器的单个Lazer-McKenna悬索桥方程的精确可控制性问题。事实证明，当仔细选择执行器的位置时，控制系统是完全可控的。精确可控性的证明是基于为非谐傅立叶级数建立Ingham不等式。
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