The object of study is the process of mathematical modeling and optimization of the structure of monitoring and control systems. The purpose of the work is to develop tools for mathematical modeling and optimization of sensor coverage of an arbitrary shape in monitoring and control systems to reduce capital investment in equipment of such systems. Research methods are based on the use of functional analysis elements to simulate the sensor zones of fire detectors; method of phi-functions for formalization of relations between geometric objects in coating problems; analytical geometry for formalizing the criterion of coverage and constructing membership functions; geometric design methods for constructing mathematical models on nand the methods of non-linear and undifferentiated optimization for solving them. The practical significance of the obtained results lies in the development and implementation of the Vesta program complex to solve the optimization problem of covering an area of arbitrary spatial shape by circles of equal radius whose centers are connected by a network (including taking into account the errors of the original data) and a number of practical problems, including the correction of unacceptable coatings, minimization of the radius of cover circles, optimization of the quality of coatings etc. The created software complex can be directly applied during designing of systems of diagnostics, observation and control, during carrying out fire-technical examinations, which allows to take into account errors of the initial data, to increase the efficiency of the equipment use . The scientific novelty of the work is this: for the first time features of describing the relationship of coverage between circles of equal radius and new membership functions are proposed to describe the relations between the circles involved in the coverage and an arbitrary area of coverage that allow to describe in an analytical form the conditions of coverage of the area of circles and, unlike existing analogues, they do not require the introduction of auxiliary variables; for the first time, a generalized mathematical model for the optimization problem of covering identical circles is constructed, which, unlike existing models, preserves new analytical expressions for describing the relations between the circles of equal radius and the new membership functions for describing the relations between the circles involved in the coverage coverage area and allows to obtain locally optimal solutions to a number of important practical problems, including problems covering the area with circles that are connected by a network; the strategy of solving the optimization problem of covering an area of arbitrary shape by circles of equal radius has been improved by adding a new stage of generation of the constraint system and the objective function based on the analysis obtained by the proposed approximate starting point methods, taking into account the specific implementation of the generalized problem model; methods of interval geometry have been further developed in the form of new modeling tools for describing interval relationships between circular objects that participate in a coating, which, unlike existing analogues, can take into account the problem of covering the error of the initial information about the actual radii zones of sensory control of sensors of circles; acquired locally-optimal methods of coating construction, which allowed, unlike existing approaches, to obtain solutions for the basic realizations of the generalized problem of optimal coverage in automatic mode; Mathematical modeling of sensor area of sensor in monitoring systems has been further developed to transform physical information about sensor zones into geometric for later use in the construction of mathematical models of coating problems, which, unlike existing ones, allow to take into account the features of functioning of monitoring systems and controls. The specified software complex is used in the department of normative-technical work and control over fire protection systems and licensing of the Main Directorate of the State Service of Ukraine for Emergency Situations in Kharkiv region during the consideration of projects of automatic fire protection systems, LLC "NPP Brand" (Kyiv) for simplification design of systems of automatic fire protection, optimization of costs for equipment of objects by such systems, as well as in the educational process at the National University of Civil Ukraine's defense in teaching the subjects "Mathematical Modeling and Optimization of Security Systems" and "Fire and Production Automation", which is confirmed by the acts of implementation, which is confirmed by the acts of implementation.
