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

该建议旨在通过发展新的数学方法来加深对图上双曲系统的理解。非线性双曲方程组的耦合将是建模、分析和控制的核心。在这一领域的进展，不仅可以解决目前的数学模型中的不一致，但也提出了一种新的方法来获得耦合条件和控制机制的一般框架。从长远的角度来看，在这个方向上的适当的结果可能会允许在未来甚至进一步缩小差距之间的良好建立的理论的双曲方程在一维和多维方程。我们感兴趣的是在物理或技术描述过程中出现的问题，如生产，交通流量或流体动力学。迄今为止尚未开发的基本机制是对这些过程的现有和充分理解的微观描述。通常，微观尺度被认为是不值得或可能的模拟真实的描述，由于其计算的复杂性。在这里，我们想利用这种精细的尺度来获得新的和物理上正确的条件和控制。这是相反的先验强加的条件，目前的最先进的。此外，我们有助于双曲系统的闭环控制策略的新领域。在这里，我们认为，特别是稳定问题的流动模式，通过一种新的组合，微观和宏观的描述，以及通过博弈论的考虑。对于一般的一维和耦合双曲问题，我们开发了一个新的李雅普诺夫函数的方法来获得有效的闭环控制配方。这个问题的研究在理论和数值水平。特别是，对于生产和交通流的数学模型，在开发合适的控制配方时，还需要考虑相互竞争的个体决策。我们提出了一种基于双曲方程和博弈论的描述相结合的方法。我们调查所得的控制算法，并与现有的方法进行比较。总体而言，应在数学建模、分析和数值分析方面取得重大进展。主要的重点将是生产模型提出了一个新的领域与微观和宏观层面的描述。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.