This dissertation is concerned with stabilization and explicit energy decay estimates for a class of bilinear system and nonlinear ordinary differential equations, used to model oscillatory phenomena in electrical or civil engineering. We also focus specifically on the so-called critical homogeneous bilinear systems (HBLS) with viscous quadratic feedback for whom we compute the energy decay estimates.; Chapter 1 is an introduction, which provides a brief review of the mathematical model of an electric power system, motivating the present work and a overview of the main problem.; In Chapter 2 we consider a well-known Lienard's system of ordinary differential equations, modeling oscillatory phenomena in various applications in engineering. It is known that such a system is asymptotically stable when a linear viscous damping with constant gain is engaged. However, in many applications it seems more realistic that the aforementioned damping is not constant and does depend upon on the deviation from the equilibrium. In this Chapter we consider a nonlinear feedback, introduced in [10], which is proportional to the square of such deviation and develop a methodology using geometric "visualization" technique to derive explicit energy decay estimates for solutions of the corresponding "damped" Lienard's system.; In Chapter 3 we consider a model of a power system which consists of a single generator connected to an infinite bus by a transmission line. This model is widely used to study the effects of various controllers on power system stability. In this Chapter we introduce a smooth nonlinear stabilizing feedback, motivated by so called facts FACTS (Flexible a.c. Transmission Systems) devices, which changes the admittance of the transmission line. We apply the methodology developed in Chapter 2 to derive explicit energy decay estimates for solutions of the corresponding system of nonlinear ordinary differential equations.; Finally, in Chapter 4 we consider a Homogeneous Bilinear System (HBLS) such that the drift term A is n x n skew-symmetric matrix and provide a detailed proof of global asymptotic stability for the given system. This result is a special case of a Theorem obtained by Jurdjevic and Quinn in [10]. We also obtain explicit energy decay estimates for the given HBLS by assuming that the input matrix B is symmetric positive definite.
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机译：本文涉及一类双线性系统和非线性常微分方程的稳定和显式能量衰减估计，用于对电气或土木工程中的振动现象进行建模。我们还特别关注具有粘性二次反馈的所谓的临界均质双线性系统（HBLS），我们将为其计算能量衰减估计值。第1章是绪论，它简要回顾了电力系统的数学模型，激发了本文的工作并概述了主要问题。在第2章中，我们考虑了著名的Lienard常微分方程组，它在工程中的各种应用中对振动现象进行建模。已知当采用具有恒定增益的线性粘性阻尼时，这种系统是渐近稳定的。然而，在许多应用中，上述阻尼不是恒定的并且确实取决于与平衡的偏差，这似乎更为现实。在本章中，我们考虑在[10]中引入的非线性反馈，该非线性反馈与此类偏差的平方成比例，并开发了一种使用几何“可视化”技术的方法，以得出用于相应“阻尼” Lienard系统解的显式能量衰减估计。 。;在第3章中，我们考虑了一个电力系统模型，该模型由通过传输线连接到无限母线的单个发电机组成。该模型被广泛用于研究各种控制器对电力系统稳定性的影响。在本章中，我们介绍一种平滑的非线性稳定反馈，其受所谓的事实FACTS（灵活的交流传输系统）设备的驱动，它会改变传输线的导纳。我们应用第2章中开发的方法来为相应的非线性常微分方程组的解导出明确的能量衰减估计。最后，在第4章中，我们考虑了同质双线性系统（HBLS），使得漂移项A为n x n斜对称矩阵，并为给定系统提供了全局渐近稳定性的详细证明。这个结果是Jurdjevic和Quinn在[10]中获得的一个定理的特例。我们还通过假设输入矩阵B是对称正定的来获得给定HBLS的明确能量衰减估计。
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