DEVELOPMENT OF A GENERALIZED MODEL FOR THE PROTECTION OF A CRITICAL INFRASTRUCTURE OBJECT FROM THE DESTRUCTIVE IMPACT OF AIR ATTACK MEANS

Purpose: development of a generalized model for the protection of critical infrastructure objects from the destructive action of an air attack. Theoretical framework: based on the analysis of the use of cruise missiles with radar correlation-extreme algorithms to damage critical infrastructure objects in the conditions of the Russian-Ukrainian war, a generalized model of the protection of critical infrastructure objects from the destructive action of an air attack has been developed. Methods: determined by a set of solved scientific and research tasks and carried out using: methods of system analysis - during studies of the distribution function of the electronic subsystem; numerical modeling methods - when studying the main electrophysical properties of critical infrastructure objects. Results and conclusions: a generalized model of the protection of a critical infrastructure facility against the destructive action of air attack tools has been developed, which allows for the assessment of risks and their management. The developed model can quantify uncertainties, simulate potential scenarios and assess the impact of various factors on the level of risk. This allows decision makers to make informed choices and develop strategies to mitigate risks.


INTRODUCTION
The first mass-produced cruise missile was created during the Second World War in Germany -"Fau-1". One of the main difficulties of the development was the guidance system. It looked as simple as possible -the autopilot monitors the course and altitude, measures the flight range. As soon as the "Fau-1" flew the specified distance, the autopilot directed the missile into a dive. Later, the Germans developed the first inertial guidance system based on analog instruments with a gyroscope and accelerometer for the Fau-2 ballistic missile (Christopher, 2013, p.26). Therefore, it was quite logical to invent a mechanism that would regulate the indicators. TERCOM became this system for cruise missiles that had to fly 3 hundreds of kilometers and stay in the air for hours. Its principle is that the missile scans the surface below it and compares it with a standard. One of the simplest implementations will be, for example, terrain data -elevation differences recorded by a radio altimeter. In this case, the missile route can be divided into certain control points, the map of which is stored in the missile's memory. These should be areas with contrasting topography, for example, rivers with steep banks, a network of ravines, or even individual large buildings. Having arrived, the rocket compares, finds and corrects the indicators of the inertial system, resetting the accumulated error. Currently, the TERKOM system uses not only terrain data, but also visual images. This is due to the fact that the route may not have a characteristic relief. But this system is much more complex because it requires work at the level of pattern recognition, but this technology, called DSMAC, was successfully mastered in the United States in the 1980s and integrated into the Tomahawk Block II cruise missile. The advent of satellite navigation fundamentally changed the situation, because now it became possible to constantly receive one's coordinates, altitude and speed. It was for this that the US initially began to deploy the GPS system in the 1970s, and in the 1980s the USSR began to deploy its GLONASS. Regarding the use of cruise missiles by the Russian Federation, the situation is as follows. Aviation X-101 and X-555 have all four components of navigation. Caliber missiles most likely do not have DSMAC (Dementiiuk et al., 2023, p. 29-37). But in the realities of the Russian Federation, another important factor is the availability of detailed, up-to-date and accurate radar and optical maps that are loaded into the missile's memory. Now the enemy is increasingly using Kh-59M missiles, the most common of which, most likely, use only inertial and satellite navigation, and for direct guidance, a radar or television guidance system is included (Kozubenko O. & Shulman O., 2022).
As for the Kh-22, it generally only has an inertial on the march section and a radar guidance system on the terminal. Both with extremely low accuracy on the Soviet technology base of the 1960s and 1970s. That is, it is launched in the direction of the target, which, moreover, should be as radiocontrast as possible. Accordingly, the Kh-22 is capable of causing much greater destruction than most modern Russian missiles. But it has a significant drawback -low accuracy (Kozubenko O. & Shulman O., 2022).
The highest accuracy was achieved in the mode of active operation of the homing head on the entire flight path (Turinskyi et al., 2019, p. 542-548). But in this mode, the missile becomes visible to air defense systems at a long distance. Most likely, Russia uses a combined guidance mode: the missile flies autonomously for most of the flight, and only at a certain distance from the target does the homing head turn on. In this mode, the accuracy drops significantly and is several hundred meters, but the chances of interception decrease. On December 1, 2022, at a briefing of representatives of the Security and Defense Forces of Ukraine, fragments of the combat part of the Kh-55SM missile, which Russia uses during shelling of Ukraine, were demonstrated (Skoblikov O. & Knyazyev, 2012, p. 1-8.). This is a modification with an increased range of the Soviet Kh-55 cruise missile, with which Russia will attack Ukraine from March 2022 from Tu-95 and Tu-160 strategic bombers. The Kh-55 and Kh-555 missiles fly at subsonic speeds with terrain at an extremely low altitude. They are intended for use at stationary strategically important objects. With the beginning of hostilities, information appeared about the first use of Kh-59 missiles. It's an old Soviet rocket from the 1980s, but it's pretty accurate. The possible circular deviation is indicated as less than 10 m, but the USSR and Russia tend to exaggerate the real accuracy of their weapons. In order to terrorize and intimidate Ukrainians, Russia strikes almost every day with the help of cruise missiles, in particular Kalibr, which are launched both from Iskander operational-tactical air systems and from ships. Launches are carried out beyond the range of Ukrainian weapons (Kozubenko O. & Shulman O., 2022).
There is nothing revolutionary in the Kalibr cruise missile, it is an updated version of the Soviet-developed 3M10 missile, which in turn was a tracing paper from the American Tomahawk cruise missile. The Soviet Kh-55 and its more modern modification Kh-555 became an alternative to long-range Kalibr missiles. But these missiles can no longer be called highly accurate. For them, the circular slope is 20-100 m.
Russia uses P-800 Onyx cruise missiles to strike targets in southern Ukraine. This missile was developed in the late 1970s as a medium-range anti-ship missile. A missile with a reduced flight range (300 km versus 600 km) is exported under the name "Yakhont". In the Russian-Ukrainian war, the Kh-101 is used -the latest cruise missile, which is launched from the Tu-160 and Tu-95MS missile carriers. It is difficult to detect, intercept and shoot down by means of air defense (Datsenko, 2022). The peculiarity of this cruise missile is that it is able to change the target even in flight.
A large number of works were devoted to the development of methods and means of passive protection of objects, which were carried out and are carried out by such famous scientists as V. Gorodnov (Horodnov et al., 2004)   and others.
The existing methods and means are not able to ensure the necessary effectiveness of the protection of critical infrastructure objects due to their peculiarities, in relation to the destructive effect of cruise missiles with a radar navigation method -an insufficient number of air defense means for the protection and distribution of critical infrastructure objects (Sytenko, 1965, p. 1-183).
Therefore, a contradiction arose, which is due, on the one hand, to the presence of the destructive effect of cruise missiles with a radar guidance method, and on the other hand, to the lack of technologies, methods and means that will allow to ensure the necessary level of protection of critical infrastructure objects without harming their functioning and which can be implemented at critical infrastructure facilities without significant financial losses and the involvement of air defense resources (Iasechko М., Atamanenko I. et al., 2019, p. 614 -617).
The idea of research is aimed at increasing the level of protection of objects of critical infrastructure in the event of repeated attacks with radar correlation-extreme guidance algorithms.
The purpose of the article is to create a model for the protection of a critical infrastructure object under the conditions of the destructive action of air attack means.
The object of the research is the process of protection of technical buildings, turbine (engine) halls of a critical infrastructure object based on changing the contrast of the critical infrastructure object, using false targets, physical reflection and changing the effective scattering area of the object.
The subject of the research is the methods of protecting the critical infrastructure object from the destructive impact of air attack means.

THEORETICAL FRAMEWORK
When comparing various mathematical models that provide the calculation of the desired parameters of the protection of critical infrastructure objects, the problem of quantitative measurement of the absolute or at least the relative value of the effectiveness of the models arises. Such a task leads to the need to choose an appropriate indicator of the effectiveness of mathematical models, which quantitatively reflects the degree of achievement of the goal of modelling (Gorodnov, 1987, p. 273-284). This indicator is naturally chosen based on the purpose of applying the mathematical model. Usually, the goal of modeling the protection of a critical infrastructure object is to optimize actions to protect the object, increase its readiness and effectiveness of cover, that is, increase the effectiveness of individual elements of protection. Then, from the point of view of the protection of the critical infrastructure object, the model that is used should provide an increase in the effectiveness of the cover due to the optimization of the protection parameters. If we are talking about optimal parameters (that is, the best in the given content), then any deviation from the optimal values of the protection parameters will lead to a decrease in effectiveness, that is, to losses in the effectiveness of the cover. Therefore, the better the model, the smaller the a posteriori loss of the effectiveness of the protection of the critical infrastructure object it provides. Then the ideal model should provide minimal Пеі efficiency losses caused by errors in the input data of the model. Taking into account the above-mentioned understandings, to compare the quality (efficiency) of two models -the evaluated and the existing (available) one, it is advisable to introduce a dimensionless (relative) indicator of the effectiveness of the evaluated model of the species: Where, Пен, Пеd, Пео are the expected loss of effectiveness of the protection of the critical infrastructure object with the direct implementation of the protection parameters formed using the existing, ideal model and the one being evaluated, respectively.
When calculating the values of such an indicator, the units of measurement of the effectiveness of the protection of the critical infrastructure object will be insignificant, and the errors in the estimations of the losses of efficiency Пе for the analyzed models will tend to mutual compensation. It can be shown that the values of the efficiency indicator lie in the range from a negative infinite value to unity (because the loss of efficiency of the protection of the critical infrastructure object when using the ideal model by its definition will be the smallest possible for the case of using any other model).
In this sense, the indicator is satisfactory. However, its direct measurement is hardly possible. Therefore, it is necessary to find the possibility of its calculation based on the results of measurement or calculation of side (indirect) parameters of the model, which directly affect the quality of the solution to the protection optimization problem. If we do not touch the methods and methods directly used in modeling, then the main quality parameters of the model, which are directly or indirectly measured, can be chosen as the known reliability of calculations, operational efficiency of modeling, completeness and importance of input data used to obtain the result (that is, taken into account in the model).
For the following considerations, it is necessary to make a number of basic assumptions. First, let us assume that the various efficiency losses Пе*, which are determined by the inaccuracy of determining each of the Q (і=1...,Q) protection parameters, are independent and additive from the point of view of the overall efficiency losses, which determine the quality of the models (* -index "n", "d", "o" of the corresponding model), i.e (2) Secondly, we will introduce the notation for the losses Пбі of the effectiveness of the cover of the critical infrastructure object, when the simulation results were not used for some reason in the protection of the object (without using the model), and for the losses of the effectiveness of the cover Пм*і, obtained when the results of the simulation and relevant 6 recommendations were used. We denote by P* -the probability of obtaining simulation results in time using this model for the time t < tн, which is available. Then the loss of effectiveness Пе*і, for each of the protection parameters, can be estimated as a mathematical expectation of the loss of effectiveness of covering the critical infrastructure object ( ) Third, suppose that each of the models (estimated, available, and ideal) provide the determination of all of the Q parameters that are sought, but the efficiency of determining each parameter in the general case is different. Then for each i-th parameter (i = 1,...,Q) for these models it will be permissible to write Taking into account the fact that the reduction in the effectiveness of the cover due to the failure to use the ideal model is always greater than due to the failure to use any other, that is, obvious inequalities Sdi  Soi; Sdi Sнi , let's move on to determining the relative values of the reduction in the loss of cover efficiency due to the non-use of the appropriate models: Where, the value of R will be determined on the interval: If we set the relative weight of the efficiency gains provided by each іth (і = 1, ... , Q) protection parameter, in the form (7) After the numerator and denominator of the expression is divided by the sum of all values of the reduction in the effectiveness of the cover Sdі at і = (1, ... , Q), the desired model efficiency indicator will take the form:

(8)
It is obvious that the expression аi·Роi·Роi simultaneously characterizes the reliability and efficiency provided by the evaluated model when calculating the i-th protection parameter, as well as the importance of this parameter, which ultimately determines the contribution of the evaluated model to this i-th protection parameter in reducing losses the effectiveness of the cover in comparison with the decision-making situation without the use of the estimated model.
Then the value (9) will approximately characterize the contribution of the analyzed (*) model to the reduction of efficiency losses for all Q protection parameters and, thus, makes sense of the degree of expected completeness of modeling for the analyzed (*) model.
Taking into account the above considerations, the indicator for comparative assessment of the effectiveness of models can be written in its general form: Tcp time -can have an interpretation of the average value of the time required for simulation (calculation) and obtaining the result.
In the case when the value of Tcp takes a constant value of T, the expression takes a simpler form: Thus, the probability P(t) essentially determines the efficiency of obtaining a result with known restrictions on the available time t and the required time T for modeling or calculations.
Due to the fact that an increase in the degree of adequacy of the model and its approximation to the ideal model is accompanied by a decrease in the absolute value of the methodological error δmet (or its dispersion -Dмет), as a quantitative indicator of the degree of inadequacy of the real model, it is appropriate to take the ratio of the form: In the general case, as the degree of adequacy of the model to the real process increases, 8 the value of this indicator approaches zero, or on the contrary, increases with the increase of inadequacy of the model. In a number of research works, including those carried out under the leadership of V. Horodnov (Gorodnov, 1987, p. 289-296), it is shown that the value of the relative error βj (j=1,2,3,4) depending on the method of accounting for factors is usually within the following limits: β1 = 0 when the factor is directly taken into account by setting its current value, which corresponds to the value in the real process; β2 = 0.4 .. 0.49 -with a simple generalization (replacement of a set of various but homogeneous in physical content factors by one factor); β3 = 0.6 -with functional and conceptual generalization of disparate factors with the aim of displaying them in the model with one representative value; β4 = 1.0 -with indirect or implicit consideration of factors.
Knowing the relative аі weights of significant factors and methods of their generalization in the model allows, after sufficiently complex mathematical transformations, to directly determine the value of the indicator of inadequacy of the model to the real process: (13) Where, Q -is the value of the model inadequacy indicator; ai -weights of the importance of taking into account the i-th factor in the model in relative units; qj -a set of factors taken into account in the model by the jth method of generalization; βj -the relative average value of the error introduced into the calculations due to inaccurate (generalized) consideration of factors. the value of the R model reliability indicator will take the form: Where, * k R -the reliability value of the definition of the k-th parameter; i -the importance of taking into account the i-th factor in the model; * jk g -a set of factors that are taken into account by the j-th method of generalization; j -the relative average value of the error that is introduced into the calculations due to inaccurate (generalized) consideration of factors Thus, a concise method of practical calculation of the effectiveness of mathematical models is reduced to the following.
From the beginning, each of the Q sought protection parameters is determined and characterized by its importance ak (priority when making a decision), the efficiency P*(t) of calculating its value using the appropriate model (usually all Q parameters have the same value of the efficiency indicator if calculated using the same model) and reliability R*k . The symbol (*) takes the value of the model number. 9 Then further, taking into account the need to have an estimate of the values of all parameters before the decision is made, the value is calculated which roughly characterizes the contribution of the considered model to the reduction of efficiency losses for all parameters sought, and thus has a sense of the degree of its expected completeness.
The generalized indicator of the effectiveness of the model has the form Where, Y1(t), Y2(t) is the expected completeness of consideration of significant factors when using the first, for example, the existing model and the second, for example, the developed model.

METHODOLOGY
The article uses the method of system analysis and the method of mathematical modeling. The method of system analysis is used to study, evaluate and understand the complex system of protection of a critical infrastructure facility. It involves breaking down the system into its components, studying their relationships, and analyzing how they function together to protect it from air attack. This method aims to identify problems, deficiencies or opportunities for system improvement, and to propose solutions or improvements. The method of mathematical modeling is used for the analysis and research of the protection of a critical infrastructure facility against an enemy air attack. It is a technique used to model, analyze and predict the behavior and results of a critical infrastructure object protection system using mathematical principles and formulas. The given mathematical model captures and describes their interaction and influence on each other.

RESULTS AND DISCUSSION
It is important to create a mathematical model for assessing the probability of providing protection of critical infrastructure objects from air enemy strikes. Mathematical models make it possible to systematically analyze the risks associated with the protection of critical infrastructure objects from strikes by an aerial enemy. Taking into account factors such as geographic location, potential threats and infrastructure vulnerabilities, the model helps estimate the likelihood and potential impact of such events.      Physical security measures (including technologies) are used to counter air attack means at critical infrastructure facilities (Iasechko, 2017, p. 18-21). This is protection when protection requires a multi-level of different measures. The basic principle is that the security of the infrastructure is not significantly impaired by the loss of any individual layer. To detect any unauthorized access and mitigate the threat before it can reach core facilities, a multi-layered approach can include the following: • delineation of the perimeters of the critical infrastructure facility area and protection by physical barriers; • patrolling and sufficient supervision; • access control with additional security features used to increase its performance or efficiency; • use of such technologies as methods and/or techniques of verification; Physical security measures must be supported by properly trained personnel, robust and reliable comprehensive emergency planning, and concise, well-written security plans and orders.Thus, the mathematical model for estimating the probability of providing cover for critical infrastructure objects from air enemy strikes is important for risk analysis, cost estimation, and resource allocation. This enables understanding and managing the risks associated with such events, supporting the resilience and recovery of critical infrastructure in challenging circumstances.

CONCLUSIONS
The research carried out in the article makes it possible to develop a mathematical model for ensuring the protection of critical infrastructure objects from the destructive impact of various types of air attack means (Iasechko М., Kolmykov M. et al., 2020, p. 1380-1384. A mathematical model can be used to predict future events or outcomes. By analyzing input data and using mathematical techniques, the model can project trends, estimate probabilities, and provide valuable information for decision-making. The mathematical model of the protection of the object of critical infrastructure from the destructive influence of air attack means allows modeling and analysis of events without the need for conducting expensive or lengthy physical experiments (Iasechko M., Larin V. et al., 2019, p. 3566 -3571). Using mathematical equations and computational tools, we can explore a wide range of possibilities,  14 saving resources and speeding up the decision-making process (Nikoliuk et. al, 2023;Zelenin, 2023). In general, a mathematical model of the protection of a critical infrastructure object from the destructive effects of air attack provides a powerful basis for understanding, predicting and optimizing air defense, contributing to progress in various fields and enabling evidence-based decision-making.