Parametric optimization of the continuous medium transferring process over a network carrier

The article considers the issue of optimal impact on the continuous medium transferring process over a network carrier, which exerts its influence on the process at the nodal points (branch points) of the network. The mathematical model of this process is defined by the formalisms of the initial-boundary value problem for a differential equation of parabolic type with distributed parameters on a graph (hereinafter referred to as a differential system on a graph). The optimizing function (in the studies of foreign researchers, for example, in the works of J.-L. Lions, the cost function) is specified by a functionality within a limited set of the admissible parameter changes space, which is a range of functions aggregated by a spatial variable. The analysis of the outlined objective is carried out by reducing a differential system to a differential-difference system, using the semi-discretization method with respect to a time variable (an equivalent of the E. Rote method); besides, the differential-difference system inherits the basic properties of the original one: unique solvability and continuity with accordance to the initial data. Thus, the mathematical model of the transfer process under study is determined by a differential-difference system with an error in the time variable proportional to the sampling step. The possibility of reducing the error to the one which would be proportional to the square of the sampling step is also indicated. The latter is dictated by the need to algorithmize as efficiently as possible the search for the optimal set of the impact parameters on the differential system, and therefore, on the studied process of continuous media transfer. The study thoroughly employs the conjugate state and the adjoint system for a differential-difference system, in which terms the relations that ascertain the optimal set of parameters are obtained. An algorithm for finding this set is given. The achieved results underlie the analysis of other optimization problems for the processes of continuous media transfer while revealing interesting parallels with multiphase problems of multidimensional hydrodynamics.

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