Solutions stability of autonomous systems of differential equations in research of wagon flow control systems for the movement of wagons to port railway stations

   The workload of the main railway directions and the need to manage railway wagon flows in conditions of limited throughput abilities of transport infrastructure accelerate the development and use of new information control information tools. As a result, there is a change in the transportation control paradigm, including through the delegation to intelligent systems of some of the functions traditionally performed by the dispatch apparatus. Thus, the number of control levels and the relationship between them changes. These issues initiate the development of new approaches and mathematical methods in transport and logistics research, in particular, in areas related to transport systems.
   The researches methodology of the organizational and technological sustainability of traffic control systems in railway transport is represented by “hard” and “soft” mathematical models, which are described by autonomous systems of ordinary linear differential equations. The transition from “hard” models to “soft” models is carried out by disaggregating the former. This is accomplished by introducing additional connections between the subjects of the considered two-level transportation process control system.
   The mathematical apparatus of the research is represented by the phase space (more precisely, the phase plane). The behavior of the trajectory corresponding to the unperturbed solution (equilibrium position) and the trajectories corresponding to the perturbed solutions of the system of differential equations under consideration is researched. Classical methods have shown that in this situation the system equilibrium point on the phase plane is the focus. Thus, it was established that for the «soft» model under consideration, the position of equilibrium is not only stable in Lyapunov, but also asymptotically stable. A phase portrait of a system of differential equations has been built, which describes the «soft» model. A comparative analysis of the geometric picture of the behavior of phase trajectories with graphical representations of the function, which expresses the daily number of wagons sent to the station, was performed.
   These results will be in demand when developing intellectualized control systems for train and wagon flows, since they are a mathematical justification for the stability of the functioning of these systems.

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