Among the major achievements of modern mathematics in one of the first places should be mentioned the mathematical theory of optimal control. It exists in two aspects: continuous and discrete .. A continuous version of the theory, which studies controlled objects described by differential equations, is known to the reader from a number of detailed monographs. At the same time, a discrete version of the theory, no less important in theoretical terms and in applications, is nowhere fully stated. The book fills this gap in the domestic and foreign mathematical and technical literature . The mathematical theory of optimal control for discrete-time objects is presented in a form available to an engineer who has mathematical training in the volume of the abscess. The presentation includes new methods and results, so that the book is interesting and the reader-mathematics . For the convenience of the reader, the book is divided into five chapters, each of which is a separate complete whole. A more detailed description of the book's chapters is given in the foreword.