Optimization of a discrete-time system for transferring a continuous medium over a network carrier

The technologies for transferring continuous media (gas, oil, petroleum products, etc) use carriers (main pipelines) with a topological structure similar to that of a geometrical graph. A large volume of literature is devoted to the issues of mathematical modeling of transfer processes along such carriers as well as to the analysis of various kinds of optimization problems related to them, but the mathematical justification of the findings is not sufficient from the standpoint of the general mathematical theory of heat and mass transfer. The paper considers the problem of a differential-difference system optimization, which determines the discrete-time equivalent of a differential system for the transport equation on a graph (in applications, on a network). E. Rote's method is employed, which is based on semi-discretization with respect to the time variable of the initial-boundary value problem, which helps to establish not only the conditions for the solvability of the specified problem, but also to obtain an optimization problem for the differential-difference system. Moreover, the coercive property of the elliptic operator bilinear differential form and the continuity of the quadratic functional being minimized are necessary and sufficient conditions for the unique solvability of the optimization problem. The findings are applicable in modeling network-like processes of continuum transport by formalisms of differential-difference systems with a spatial variable fluctuating on a network-like multidimensional domain. The conditions that determine the solution of the optimization problem or the set of such solutions are presented. Concurrently, approaches to the analysis of the optimization problem for a system defined on a multidimensional network-like domain are outlined. The findings underlie the analysis of optimal control problems for differential systems with distributed parameters on a graph, which have interesting analogies with multiphase problems of multidimensional hydrodynamics.

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