Russian Mathematics (Iz. VUZ) Vol. 47, No. 12, pp.1{27, 2003 Izvestiya VUZ. Matematika UDC 517.977 OPTIMIZATION OF DYNAMIC SYSTEMS IN THE CLASS OF DISCRETE CONTROLS OF FINITE DEGREE R. Gabasov, F.M. Kirillova and N.S. Pavlenok In this paper, we consider the problems of optimal control (OC) of linear systems in the class of discrete controls which are supposed to be representable as polynomials of nite degree on each time quantization period. Taking into account the geometrical restrictions imposed on the coecients of these polynomials and on the control variables for each instant of time, we formulate the problems mentioned above as complex extremum problems. We propose techniques for calculating the optimal programs and synthesizing the feedback type OC for two classes of control actions, namely, for the rst degree discrete controls with the nite number of time quantization periods and the nite degree controls with one time quantization period. These techniques are based on the dynamic implementation of one particular linear programming (LP) technique. The solutions of the OC problems are used for the synthesis of the limitary stabilizing feedbacks according to the moving control principle. The techniques are illustrated with the numeric examples. 1. Introduction The mathematical theory of optimal processes 1] is based on rather broad classes of controls which include, in particular, the measurable and piecewise continuous functions. In the constructive OC theory, which is aimed on the computer-aided problems solution, the more specic classes of controls are used, such as discrete controls. The latter may change only at the predened points of time and keep the constant values on the intervals between these points. In this paper, we call such controls the discrete zero degree controls. Their natural generalizations are the discrete nite degree controls. In general, the discrete nite degree controls with the constant quantization period are representable as follows: u(t) = p X j =0 uj (tk )'j (t ; tk ) t 2 tk tk+1 k = 0 N ; 1 where tk = t + kh, h = (t ; t )=N 'j (t), t 2 0 h, j = 0 p are the given linearly independent functions, uj (tk ), j = 0 p, k = 0 N ; 1 are the parameters of the discrete function. We choose the basic functions 'j (t), t 2 0 h, j = 0 p due to the specic features of the OC problem under consideration. Mostly, we use the following simplest type of the basic functions 'j (t) = tj , j = 0 p. p The set of corresponding controls we denote by UN . The extension of the classical type of zero degree discrete controls is also useful for the qualitative control theory. It is well known that the investigation of the continuous control system x_ = Ax + bu (A 2 Rnn b 2 Rn) (1) The work was supported by the Byelorussian Foundation for Basic Research (projects nos. F03M-031, F02R-008) and the Byelorussian government program of Basic Research \Mathematical Structures". c 2003 by Allerton Press, Inc. Authorization to photocopy individual items for internal or personal use, or the internal or personal use of specic clients, is granted by Allerton Press, Inc. for libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the base fee of $ 50.00 per copy is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923. 1