Qualitative Analysis of Complex Dynamical Systems

在过去的十九年里，首席研究员、他的同事和学生在复杂动力系统的定性分析方面进行了研究。这方面的大量工作涉及到Lyapunov稳定性、Lagrange稳定性、输入输出性质、适定性、周期运动的存在性和稳定性、分解、拓扑性质以及互联动力系统的选择应用。他建议沿着以下方向继续他对复杂动力系统的研究：1.研究混合动力系统的定性性质，重点是数字动态控制器在数字反馈控制系统中的实现。2.利用分解技术研究了无限维动力系统的Lyapunov稳定性(例如，变换成以强连通分量为子系统的块下三角型系统)，并建议在这项工作中强调有毒废物模型。3.将神经网络视为相互关联的动力系统，研究了几种神经网络模型的定性性质和综合方法。4.利用非保守动力系统类哈密顿变分方法的最新结果，研究了包含传递电导的电力系统模型；他的目的是为现有的几个利用这类系统的路径相关能量函数的启发式结果提供分析依据。5.继续研究复杂动力系统中周期运动的存在性、不存在性、稳定性和不稳定性。他对严格建立各种描述函数方法的有效性特别感兴趣。6.将区间分析技术应用于动力系统的研究。重点将放在参数公差问题上。7.在建立对象模型的基础上，利用分数时间导数研究受控粘弹性结构的稳定性。这些课题显然在理论研究和应用中都具有重要意义。他的方法论将涉及泛函分析技术、弗洛奎特理论的结果、微分方程组的不变性和比较理论、李雅普诺夫稳定性理论(对于有限和无限维系统)、I/O稳定性理论、区间分析等。

During the last nineteen years, the principal investigator, his co- workers and students have conducted research in the qualitative analysis of complex dynamical systems. A great deal of this work is concerned with Lyapunov stability, Lagrange stability, input-output properties, well-posedness, existence and stability of periodic motions, decomposition, topological properties, and selected applications of interconnected dynamical systems. He proposes to continue his investigations of complex dynamical systems along the following lines: 1. Study qualitative properties of hybrid dynamical systems with an emphasis on digital implementations of dynamic controllers in digital feedback control systems. 2. Study the Lyapunov stability of infinite dimensional dynamical systems, using decomposition techniques (e.g., transformation into systems in block lower triangular form having strongly connected components as subsystems), and he proposes to emphasize toxic waste models in this work. 3. Views in neural networks as interconnected dynamical systems, study qualitative properties as well as synthesis procedures for several models of neural networks. 4. Using recent results for Hamilton-like variational methods of nonconservative dynamical systems, study power system models which include transfer conductances; he aims to provide analytical justification for several existing heuristic results which make use of path dependent energy functions for such systems. 5. Continue research on existence, nonexistence, stability and instability of periodic motions in complex dynamical systems. He is particularly interested in establishing rigorously the validity of various describing function methods. 6. Apply interval analysis techniques in the study of dynamical systems. The emphasis will be on parameter tolerance problems. 7. Study the stability of controlled viscoelastic structures using fractional time derivatives in the model formulation of the plant. These topics are clearly of great importance in theoretical studies as well as in applications. His methodology will involve functional analysis techniques, results from Floquet theory, invariance and comparison theory for differential equations, Lyapunov stability theory (for finite and infinite dimensional systems), I/O stability theory, interval analysis, etc.