Optimal boundary control of the parabolic system with distributed parameters on the graph

In this work we consider the case when the state of the differential system with distributed parameters on the graph is determined by a weak solution of the initial boundary value problem on the graph. Such solution belongs to a Sobolev space and satisfies the conditions of transversality in all internal nodes of the graph. The impact on the system and monitoring of the system's state are carried out at the border (in the boundary nodes of the graph) for an arbitrary time variable. The adjoint state of the system is determined by the weak solution of the initial boundary value problem on the graph with the final (by time) condition. The necessary and sufficient conditions for the existence of the single boundary control are obtained. All considerations are given for an arbitrary connected limited directed graph, allowing for the presence of cycles (loops).