The technology of applying the variational principle in problems of development and testing of complex technical systems is described. Let there be a certain set of restrictions imposed on random variables in the form of given statistical moments and/or in the form of a restriction by some estimates from above and below the range of possible values of these random variables. The task is set: without knowing anything except these restrictions, to construct for further research, ultimately, for assessing the efficiency of the complex technical system being developed, the probability distribution function of its determining parameter. By varying the functional, including Shannon entropy and typical restrictions on the distribution density function of the determining parameter of a complex technical system, the main stages of constructing the distribution density function are described. It is shown that, depending on the type of restriction, the constructed distribution density function can have an analytical form, be expressed through special mathematical functions, or be calculated numerically. Examples of applying the variational principle to find the distribution density function are given. It is demonstrated that the variational principle allows obtaining both the distribution laws widely used in probability theory and mathematical statistics, and specific distributions characteristic of the problems of developing and testing complex technical systems. The technology of applying the variational principle presented in the article can be used in the model of managing the self-diagnostics process of intelligent control systems with machine consciousness.

Keywords: variational principle, distribution density function, Shannon entropy, complex technical system

The current situation in the practice of designing complex technical systems with metrological support is characterized by the following important features: a) the initial information that can actually be collected and prepared at the early stages of design for solving probabilistic problems turns out, as a rule, to be incomplete, inaccurate and, to a high degree, uncertain; b) the form of specifying the initial information (in the form of constraints) in problems can be very diverse: average and dispersion characteristics or functions of them, measurement errors or functions of them, characteristics specified by a probability measure, etc. These circumstances necessitate the formulation and study of new mathematical problems of characterizing distribution laws and developing methods and algorithms for solving them, taking into account the constraints on the value and nature of change of the determining parameter (random variable) of a complex technical system. As a generalized integral characteristic of the determining parameter, the law of its distribution is chosen, which, as is commonly believed, fully characterizes the random variable under study. The purpose of this work is to develop a method that allows constructing distribution laws of the determining parameter of a complex technical system using the minimum amount of available information based on the application of Chebyshev inequalities. A method for characterizing the distribution law by the property of maximum entropy is presented, designed to model the determining parameter of complex technical systems with metrological support. Unlike the classical characterization method, the proposed method is based on the use of Chebyshev inequalities instead of restrictions on statistical moments. An algorithm for constructing the distribution function of the determining parameter is described. A comparison is given of the results of constructing distribution laws using the developed method and using the classical variational calculus.

Keywords: Chebyshev inequalities, complex technical system, design, determining parameter, characterization of distribution law