Novel models and control for networked problems: from discrete event to continuous dynamics

这项提议旨在通过发展新的数学方法来加深对图上双曲系的理解。建模、分析和控制的核心将是非线性双曲型方程的耦合。这一领域的进展不仅可以解决当前数学模型中的不一致之处，而且还可以提供一种新的方法来推导一般框架的耦合条件和控制机制。从长远来看，这方面的适当结果今后甚至可以进一步缩小一维双曲方程和多维方程之间的差距。我们对生产、交通流量或流体动力学等过程的物理或技术描述中出现的问题感兴趣。到目前为止还没有开发的基本机制是对这些过程的现有和众所周知的微观描述。通常，由于计算的复杂性，微观尺度被认为是不值得或不可能进行真实描述的。在这里，我们想要在当地利用这一精细的规模来获得新颖的和物理上正确的条件和控制。这与先验强加的条件背道而驰，这些条件呈现了当前的最先进水平。此外，我们还对双曲系统的闭环系统控制策略这一新领域做出了贡献。在这里，我们通过微观和宏观描述的新组合以及通过博弈论的考虑来特别考虑流型的稳定性问题。对于一般的一维和耦合双曲问题，我们发展了一种新的Lyapunov函数方法来推导有效的闭环系统控制公式。从理论和数值两个层面对该问题进行了研究。特别是，对于生产和交通流中的数学模型，在开发适当的控制公式时，还需要考虑相互竞争的个别决策。我们提出了一种基于双曲方程描述和博弈论相结合的方法。我们研究了所得到的控制算法，并与现有的方法进行了比较。总体而言，在数学建模、分析和数值分析方面应取得重大进展。主要焦点将放在生产模型上，呈现一个具有微观和宏观两个层面的描述的新领域。开发的新技术也将应用于交通流和能源运输系统的模型。

The proposal is intended to deepening the understanding of hyperbolic systems posed on graphs by developing novel mathematical methods. The core of the modeling, analysis and control will be on the coupling of nonlinear hyperbolic equations. Progress in this field would allow not only to resolve inconsistencies in current mathematical models, but also to present a novel method to derive coupling conditions and control mechanisms for a general framework. In a long-term perspective suitable results in this direction may allow in future even to further closing the gap between the well-established theory of hyperbolic equations in one dimension and multi-dimensional equations. We are interested in problems arising in physical or technical descriptions of processes like production, traffic flow or fluid dynamics. The basic mechanism not exploited yet so far, is the present and well-understood microscopic description of such processes. Usually, the microscopic scale is considered not worth or possible simulating for realistic description due to its computational complexity. Here, we want to exploit this fine scale locally to derive novel and physically correct conditions and controls. This is contrary to a priori imposed conditions which present the current state-of-the-art. Additionally, we contribute to the rather new field of closed loop control strategies for hyperbolic systems. Here, we consider in particular stabilization questions of flow patterns through a novel combination of microscopic and macroscopic descriptions as well as through game theoretic considerations. For general one-dimensional and coupled hyperbolic problems we develop a new Lyapunov function approach to derive efficient closed loop control formulations. The problem is studied on a theoretical and a numerical level. In particular, for mathematical models in production and traffic flow there are also competing individual decisions need to be taken into account in the development of suitable control formulations. We propose an approach based on a combination of a description using hyperbolic equations and game theory. We investigate the resulting control algorithm and compare with existing approaches. Overall, significant progress should be made on mathematical modeling, analysis and numerical analysis. The main focus will be on production models presenting a novel field with both microscopic and macroscopic levels of descriptions. The developed new techniques will also be applied to models in traffic flow and energy transportation systems.