Methods of Continuous Logic in Control

Abstract

Continuous logic, its problems, and methods were outlined, and its basic operations were defined. The algebra of continuous logic was described, and its main functions of one, two, and three variables were listed. The laws of this logic were presented and contrasted with the laws of the discrete two-valued logic. Described were the problems of listing all continuous-logic functions of a given number of variables and representing them in a standard form. The difference between these forms and their counterparts in the two-valued logic was shown. Minimization procedures for the continuous-logic functions and their decomposition into functions of a smaller number of variables were described. The distinctions of these procedures from their counterparts in the two-valued logic were noted. The problems of analysis and synthesis of the continuous-logic functions were formulated, and methods for their solution were presented. The problem of synthesis was shown not to be necessarily solvable. The fundamentals of the continuous-logic differential and integral calculuses were presented. Any continuous-logic function was shown to have no-derivative points. The problem of completeness was described for the continuous logic together with the existing results and their distinctions from the discrete case. Numerous applications of the continuous logic to mathematics, engineering, economics, social sciences, and so on were described, and its perspectives were estimated.