The flows local control problem in resource networks consists in finding a collection of capacities for arcs outcoming from a selected controlled vertex set, such that the given state \(Q'\) is a limit for any initial state \(Q^0\) . An approach to solving the out-flows local control problem in resource networks with a high resource is proposed. Two instability conditions for a given state \(Q'\) are derived. It is shown that if the conditions of instability are violated, there exists a collection of capacity values of the arcs outcoming from controlled vertices, for which the given state \(Q'\) is a limit. To solve the problem of leading initial state \(Q^0\) to stable state \(Q'\) , the method of building a pre-periodic part of dynamic resource network is developed. The criterion for the existence of a solution to the flow control problem in the pre-periodic part of the dynamic resource network, such that state \(Q^0\) is led to stable state Q, is formulated and proved.
