There are a large number of dynamic objects, for the management of which it is advisable to apply the principles of adaptation. The reasons for applying the principles of adaptation can be combined into two main groups: the variability and complexity of the characteristics of objects and the environment; growing requirements for accuracy and technical and economic characteristics of systems. The difference between adaptive systems and optimal ones is that while in optimal systems the quality indicator is provided for certain parameters of the object, in adaptive systems - for various parameters due to the action of additional elements of adaptation. The choice of one or another approach is determined by preliminary information about the object (process) or the accepted quality criterion. The article presents the main approaches to the selection of possible variants of adaptive systems with stabilization and optimization of the quality of control of systems for stabilizing dynamic objects, based on the type of extremal characteristic of the criterion for assessing their quality.