The Problem of Minimax Estimation of Functionals for Non-Stationary Diffusion Processes




minimax functionals estimation, neutron diffusion, optimal control theory, parabolic type problems


Purpose. To model the technological process of analysis of energy sources that use random interference, it is necessary to apply special methods from the theory of minimax estimation and optimal control. The article proposes a method for solving the problem of minimax estimation of functionalities for the systems with distributed parameters with incomplete data for the process of neutron diffusion in a nuclear reactor. Methodology. In practice, in the study of non-stationary controlled processes of functioning of different energy sources there are measurement errors. As a rule, the exact values of errors are unknown, and therefore the desired solution of the equations in partial derivatives describing these processes is determined ambiguously. Therefore, it is advisable to set the task of calculating such an optimal estimate, which would best approximate the unknown value, taking into account the known information about the measurement errors. The best estimate can be achieved by applying a minimax approach to estimating functionals from the solutions of the partial differential equations of parabolic type. Findings. For a mathematical model of the neutron diffusion process in a nuclear reactor, the proposed method allows solving the problem of minimax estimation of the functional determined during the solution of the system describing this process. Since in real conditions of reactor operation there are always random obstacles (both in the equation describing the process and the function observed), the method allows finding a minimax estimate of the functional. The problem is reduced to the problem of optimal control with a given quality functionality, which is successfully solved. Originality. Using the methods of minimax estimation and optimal control of systems with distributed parameters, the best a priori estimation of the quality functional of the minimax estimation problem for the mathematical model of neutron diffusion in a nuclear reactor is obtained. Practical value. The method of minimax estimation of functionalities for differential equations of parabolic type proposed in the article allows reducing the problem to the problem of optimal control of the systems with distributed parameters, which can be implemented in Maple package using known algorithms.


Babich, Y. A., & Michaylova, T. F. (2018). Approximation of Periodic Functions of Many Variables by func-tions of Smaller Number of Variables in Orlicz Metric Spaces. Ukrains’kyi Matematychnyi Zhurnal, 70(8), 1143-1148. (in Russian)

Yegorov, A. I. (1978). Optimalnoe upravlenie teplovymi i diffuzionnymi protsessami. Moscow: Nauka. (in Russian)

Yegorov, A. I. (2016). Obyknovennye differentsialnye uravneniya i sistema Maple. Moscow: COLON-PPYeSS. (in Russian)

Nakonechniy, A. G., Akimenko, V. V., & Trofimchuk, O. Yu. (2007). Model optimalnogo upravleniya sistemoy integro-differentsialnіkh uravneniy s vyrozhdayushcheysya parabolichnostyu. Cybernetics and Systems Analysis, 6, 90-102. (in Russian)

Nakonechnyy, A. G. (1985). Minimaksnoe otsenivanie funktsionalov ot resheniy variatsionnykh uravneniy v gilbertovykh prostranstvakh. Kyiv: KGU. (in Russian)

Nakonechnyy, O. G. (2004). Optymaljne keruvannja ta ocinjuvannja v rivnjannjakh iz chastynnymy pokhidnymy: navchaljnyj posibnyk. Kyiv: Vydavnycho-polighrafichnyj centr «Kyjivsjkyj universytet». (in Ukrainian)

Galaktionov, V. A., & Svirshchevskii, S. R. (2007). Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics. Chapman & Hall/CRC Press. (in English)

Kapustyan, V. O., & Pyshnograiev, I. O. (2016). Approximate Optimal Control for Parabolic–Hyperbolic Equations with Nonlocal Boundary Conditions and General Quadratic Quality Criterion (pp. 387-401). Advances in Dynamical Systems and Control. DOI: (in English)

Kapustyan, V. O., & Pyshnograiev, I. O. (2015). Minimax Estimates for Solutions of parabolic-hyperbolyc equations with Nonlocal Boundary Conditions (pp. 277-296). Continuous and Distributed Systems II. DOI: (in English)

Kogut, P. I., & Maksimenkova, Yu. A. (2017). On regularity of weak solution to one class of initial-boundary value problem with pseudo-differential operators. Journal of Optimization, Differential Equations, and Their Applications, 25(8), 70-108. (in English)




How to Cite

Mykchailova, T. F., & Maksymenkova, Y. A. (2021). The Problem of Minimax Estimation of Functionals for Non-Stationary Diffusion Processes. Science and Transport Progress, (6(96), 77–83.