The analysis of the behaviour of the evolutionary equation solution with unlimited time variable has been a subject of discussion in scientific circles for a long time. There are many practical reasons for this when the initial conditions of the equation are specified with a certain error: how the small changes in the initial conditions affect the behaviour of the solution for large values of the time. The paper uses the classical understanding of the stability of the solution of a differential equation or a system of equations that goes back to the works of A. M. Lyapunov: a solution is stable if it little changes under the small perturbations of the initial condition. In the work specified the stability conditions for the solution of an evolutionary parabolic system with distributed parameters on a graph describing the process of transfer of a continuous mass in a spatial network are indicated. The parabolic system is considered in the weak formulation: a weak solution of the system is a summable function that satisfies the integral form identity, which determines the variational formulation for the initial-boundary value problem. By going beyond the classical (smooth) solutions and addressing weak solutions of the problem the authors aim not only to describe more precisely the physical nature of the transfer processes (this takes on particular importance when studying the dynamics of multiphase media) but also to the path analysis processes in multidimensional network-like domains. The used approach is based on a priori estimates of the weak solution and the construction (the Fayedo-Galerkin method with a special basis -the system of eigenfunctions of the elliptic operator of a parabolic equation) of a weakly compact family of approximate solutions in the selected state space. The obtained results 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.
In the paper be considered the problem of optimal control of the differential-difference equation with distributed parameters on the graph in the class of summable functions. Particular attention is given to the connection of the differential-differential system with the evolutionary differential system and the search conditions in which the properties of the differential system are preserved. This connection establishes a universal method of semi- digitization by temporal variable for differential system, providing an effective tool in finding conditions of uniqueness solvability and continuity on the initial data for the differential- differential system. A priori estimates of the norms of a weak solution of differential- differential system give an opportunity to establish not only the solvability of this system but also the existence of a weak solution of the evolutionary differential system. For the differential-difference system analysis of the optimal control problem is presented, containing natural in that cases a additional study of the problem with a time lag. This essentially uses the conjugate state of the system and the conjugate system for a differential-difference system — defining ratios that determine optimal control or the set optimal controls. The work shows courses to transfer the results in case of analysis of optimal control problems in the class of functions with bearer in network-like domains. The transition from an evolutionary differential system to a differential-difference system was a natural step in the study of applied problems of the theory of the transfer of solid mediums. The obtained results 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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