Stability of differential-difference parabolic system with distributed parameters on the graph

The work presents the results of the analysis of the approximation of a differential system with distributed parameters on the network domain. This approximation consider in a space of summable functions with a compact bearer on network domain. The question is whether the differential system retains its main properties when its approximation – uniqueness weak solvability, continuity according to the input data, stability on Lyapunov and in what cases (under what conditions) it is possible to replace the analysis of the differential system by analysis of its differential-difference analogue. The study uses a simplified bearer of functions – graph, which is a private case of a network domain (the graph’s edges are parameterized by a one-dimensional spatial variable that changes on the graph). On the set of these functions study the differential-difference equation. This simplification only frees us from the unnecessary routine technical work, where the function bearer is an arbitrary the network domain of multi-dimensional Euclidean space, but as it does not affect on the transfer of ideas and results presented in the work to this multidimensional event. The transition from a differential system to a differential-difference analogue (differential-different system) is carried out using the method of finite difference on the time variable (Rote semi-digitization method). The conditions of weak uniquely solvability of differential-difference system and continuity of the input data are presented. guaranteeing the stability of a weak solution of differential-differential system. These conditions guarantee the stability of a weak solution of differential-difference system.