In many applications because of the complexity of the mathematical models have to abandon the use of ordinary differential equations in behalf of considering the evolutionary equations with partial derivatives. In addition, most commonly the evolutionary problem study on the finite interval changes of a temporary variable. In practice, where you can solve the problem for arbitrary finite interval changes to a temporary variable it is important to know the behavior of the solution where, when the temporary variable strives to infinity. First of all, this is related to the study of the properties of the stability of the indicated solution and the possibility of constructing the stabilizing control in case of the instability. Precisely this case is the object of the study in this work, in which represent the analysis of the stability of the weak solutions of the evolutionary systems with distributed parameters on the graph with the unlimited growth of the temporary variable, obtain the conditions of the stabilization of the weak solutions. By studying the relevant initial-boundary value problem, we to be beyond the scope of the classical solutions and appeal to the weak solutions of the problem, reflecting more accurately the physical essence of appearance and processes (i. e. consider the initial-boundary value problem in weak formulation). In this case, the choice of the class of weak solutions to be determined one way or the other functional space is at the disposal of the researchers and to meet the demand, above all, conservation of the existence theorems and the uniqueness theorems for the arbitrary finite interval changes to a temporary variable. The fundamental used tool is the representation of a weak solution in the form of a functional series (method Faedo - Galerkin approximation with the special basis-system functions - the eigenfunction system) and the compactness of a many of approximate solutions (thanks to a priori estimates).