Synthesis of optimal boundary control of parabolic systems with delay and distributed parameters on the graph

The problem of optimal boundary control of evolutionary systems with constant delay and distributed parameters on the graph. System status is determined by a weak solution of the boundary value problem for a parabolic equation in the space of Sobolev type whose elements are functions satisfying the conditions in a certain way matching all the internal nodes of the graph. The control action on the system and monitoring its state is made in the boundary nodes of the graph on the entire time interval. The dual status of the system is defined as a weak solution of the boundary value problem with delay and distributed parameters on the graph with the final condition. The conditions of weak unique solvability of the original and the dual challenges of weak continuous dependence of solutions on initial data. We present necessary and sufficient conditions for the existence of optimal control using the dual system state solved the problem of optimal control synthesis for the case of absence of restrictions on the control action and an analogue-known finite-dimensional case the Kalman results. The method used is applicable to many optimization problems of differential systems whose state is determined by weak solutions of evolution equations on networks. These results are fundamental in the study of problems of boundary control the dynamics of laminar flows of multiphase media. All techniques and methods can be used for the numerical solution of optimal control problem under consideration. Refs 20.