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This article solves the absolute stabilization problem for delay systems in the class of finite-dimensional controllers. The absolute stabilization problem for the class of control objects described by delay differential equations was solved in a previous study. For various applications, it is important to consider a version of this problem with the additional condition that the controller should be governed by ordinary differential equations, i.e., have a finite-dimensional state space as a dynamical system. Such controllers are usually referred to as finite-dimensional dynamical controllers.