Start control of parabolic systems with distributed parameters on the graph

The paper is raised a fairly wide range of issues, related to the theory of differential control systems is described by differential equations with distributed parameters on the graph. It is considered the common in applications the case of a start control and final observation for a differential system whose state is described by a generalized (weak) solution of the initial-boundary value problem with distributed parameters on the graph. Although the use of these methods is demonstrated for the specified control and observations, the used methods have great generality and after minor technical changes are applicable to other types of control and observation, for example the boundary. The most attention was paid to the weak unique solvability of initial-boundary value problem in different spaces and continuous dependence of weak solutions from the initial data of the problem, i. e., to the search of correctness conditions of Hadamard are determined by the function space, to which a weak solution belongs. Having sufficiently effective methods of analysis of solutions of initial-boundary value problems, it is obtained the necessary and sufficient conditions for the existence (determining) of the optimal control in terms of the relations linking the state of the system with its adjoint state. In this case, it was analyzed exhaustively the controllability of the original differential system. All of techniques and methods can be applied to the numerical solution of optimal control problems under consideration. Refs 24.