A Functional Time System Connectivity Analysis in Multi-Stage Processes

摘要

The research urgency of the problem under consideration is caused by the need to optimize discrete dynamical systems in industry, agriculture, technology, transport, economics, etc. in terms of systems theory for their sustainable development. Dynamic programming and the bang-bang principle used in solving multi-step optimization tasks in the external (physical) time do not assume a systematic approach. The goal of this article is to construct a mathematical model for optimizing multi-stage processes and to determine the harmony principle in terms of systems theory; the processes which basic representation is a symmetry group formulated by an objective function. The research method, which becomes the leading approach to solving this problem, is the revealing of symmetry group in discrete technological, energy, kinematic, economic flows accompanying multi-stage processes in the form of a multiplicative group for multiplication by a set of real numbers defined by a corresponding objective function. A set of objective functions formulate functional time periods on the basis of which a multi-stage process becomes a system in the form of a single deeply integrated structure. The bang-bang principle of multi-stage processes in the context of the functional time is reduced to synchronization of the functional time at individual stages. Therefore, the system harmony principle should be realized by superposing the bang-bang principle of multi-step processes in the external (physical) and the internal (functional) time. As a result of this research, a mathematical model of harmonious representation of technical and economic efficiency of dynamic multi-step processes in their functional time periods is given; the most informative efficiency criteria, dimensionality of functional space of accompanying flows, the bang-bang principle, etc., are formulated. The bang-bang principle of multi-stage processes in their functional time is reduced to synchronization of functional time periods of their individual stages. Therefore, the system harmony principle should be realized by superposing integrated bang-bang principles of multi-step processes in their external (physical) and internal (functional) time. The content of the article can be useful for mathematicians engaged in optimization tasks, for engineers and designers who can apply this research in their practical activities.