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

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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.