Leontief-Type Systems and Applied Problems

The article presents a set of main results obtained in recent years in analytical and numerical studies of various problems for systems of the Leontief type that is a finite-dimensional analogue of Sobolev type equations. The key factor in achieving certain success was the presence of applied problems, the study of each of which was of independent interest. The article presents three mathematical models based on the Leontief type system: a degenerate balance dynamic model of a manufacturing enterprise, a degenerate balance model of the cell cycle, and a mathematical model of a complex measuring device. As a class of problems, we consider the Schowalter-Sidorov initial problem for a Leontief-type system and a number of optimal control problems for it. Numerical methods for solving such problems are briefly outlined, and results on the convergence of approximate solutions to the exact one are shown. Particular attention is paid to the problem of restoring a dynamically distorted input signal from an observed output signal in the presence of noise. The mathematical model of a complex measuring device is constructed as a Leontief-type system, the initial state of which reflects the Showalter-Sidorov condition. The main position of the theory of optimal dynamic measurements is modelling the desired input signal as a solution to the problem of optimal control with minimization of the penalty functional, in which the discrepancy between the simulated and observed output (or observed) signal is estimated. The presence of noise at the output of the measuring device leads to the need to use digital filters in numerical algorithms. The article is of a review nature and provides an analysis of the development of research into Leontief-type systems.

Keywords

Leontief-type system; optimal control; Showalter-Sidorov condition; numerical solution algorithms; optimal dynamic measurement; degenerate balance dynamic enterprise model.

References

1. Boyarintsev Yu.E., Chistyakov V.F. Algebro-differencial'nye sistemy: metody resheniya i issledovaniya [Algebrodifferential Systems: Methods of Solution and Research]. Novosibirsk, Nauka, 1998. (in Russian)
2. Burlachko I.V., Sviridyuk G.A. An Algorithm for Solving the Cauchy Problem for Degenerate Linear Systems of Ordinary Differential Equations. Computational Mathematics and Mathematical Physics, 2003, vol. 43, no. 11, pp. 1613-1619.
3. Favini A., Sviridyuk G.A., Sagadeeva M.A. Linear Sobolev Type Equations with Relatively p-Radial Operators in Space of ``Noises''. Mediterranean Journal of Mathematics, 2016, vol. 13, no. 6, pp. 4607-4621. DOI: 10.1007/s00009-016-0765-x
4. Favini A., Sviridyuk G.A., Manakova N.A. Linear Sobolev Type Equations with Relatively p-Sectorial Operators in Space of ``Noises''. Abstract and Applied Analysis, 2016, vol. 13, pp. 1-8. DOI: 10.1155/2015/697410
5. Gliklikh Yu.E., Mashkov E.Yu. Stochastic Leontieff Type Equations in Terms of Current Velocities of the Solution II. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2016, vol. 9, no. 3, pp. 31-40. DOI: 10.14529/mmp160303
6. Hairer E., Lubich C., Roche M. The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods. Switzerland, Universite de Geneve, 1988.
7. Кeller A.V. [Leontief Type Systems: Classes of Problems with Showalter-Sidorov Initial Condition and Numerical Solutions]. The Bulletin of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no. 2, pp. 30-43. (in Russian)
8. Keller A.V. Numerical Study of Optimal Control Problems for Leontief Type Models. PhD Thesis, Chelyabinsk, 2011. (in Russian)
9. Keller A.V. The Algorithm for Solution of the Showalter-Sidorov Problem for Leontief Type Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2011, no. 4 (221), pp. 40-46.
10. Keller A.V. On the Algorithm for Solving Problems of Optimal and Rigid Control. Software and Systems, 2011, no. 3. pp. 38-42.
11. Keller A.V., Shishkina T.A. The Method of Constructing Dynamic and Static Balance Models at the Enterprise Level. Bulletin of the South Ural State University. Series: Economics and Management, 2013, vol. 7, no. 3, pp. 6-10.
12. Keller A.V., Ebel S.I. [On a Degenerate Discrete Balance Dynamic Model of the Cell Cycle]. South Ural Youth School of Mathematical Modelling, 2014, pp. 74-79. (in Russian)
13. Keller A.V. On the Computational Efficiency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type. Journal of Computational and Engineering Mathematics, 2015, vol. 2, no. 2, pp. 39-59.
14. Keller A.V. Optimal Dynamic Measurement Method Using the Savitsky-Golay Digital Filter. Differential Equations and Control Processes, 2021, no. 1, pp. 1-15.
15. Keller A.V., Sagadeeva M.A. Convergence of the Spline Method for Solving the Optimal Dynamic Measurement Problem. Journal of Physics, 2021, vol. 2021, article ID: 012074, 6 p. DOI: 10.1088/1742-6596/1864/1/012074
16. Kondyukov A.O. Generalized Model of Incompressible Viscoelastic Fluid in the Earth's Magnetic Field. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 13-21. DOI: 10.14529/mmph160302
17. Kurdjukov А.P., Belov A.A. Deskriptornye sistemy i zadachi upravleniya [Descriptor Systems and Control Problem]. Moscow, Fizmatlit, 2015.
18. Leont'ev V.V. Mezhotraslevaya ekonomika [Intersectoral Economics]. Moscow, Economics, 1997. (in Russian)
19. Manakova N.A. Mathematical Models and Optimal Control of the Filtration and Deformation Processes. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 3, pp. 5-24. DOI: 10.14529/mmp150301
20. Melnikova I.V., Alshanskiy M.A. Stochastic Equations with an Unbounded Operator Coefficient and Multiplicative Noise. Siberian Mathematical Journal, 2017, vol. 58, no. 6, pp. 1052-1066. DOI: 10.1134/S0037446617060143
21. Sagadeeva M.A., Zagrebina S.A., Manakova N.A. Optimal Control of Solutions of a Multipoint Initial-Final Problem for Non-Autonomous Evolutionary Sobolev Type Equation. Evolution Equations and Control Theory, 2019, vol. 8, no. 3, pp. 473-488. DOI: 10.3934/eect.2019023
22. Shestakov A.L., Sviridyuk G.A. A New Approach to Measurement of Dynamically Perturbed Signals. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2010, no. 16 (192), pp. 116-120.
23. Shestakov A.L., Sviridyuk G.A., Khudyakov Yu.V. Dynamic Measurements in ``Noise'' Spaces. Bulletin of the South Ural State University. Series: Computer Technologies. Control. Electronics, 2013, vol. 13, no. 2, pp. 4-11.
24. Shestakov A.L., Sviridyuk G.A., Keller A.V., Zamyshlyaeva A.A., Khudyakov Yu.V. Numerical Investigation of Optimal Dynamic Measurements. Acta IMEKO, 2018, vol. 7, no. 2, pp. 65-72. DOI: 10.21014/acta_imeko.v7i2.529
25. Shestakov A.L., Keller A.V., Zamyshlyaeva A.A., Manakova N.A., Zagrebina S.A., Sviridyuk G.A. The Optimal Measurements Theory as a New Paradigm in the Metrology. Journal of Computational and Engineering Mathematics, 2020, vol. 7, no. 1, pp. 3-23. DOI: 10.14529/jcem200101
26. Shestakov A.L., Zagrebina S.A., Manakova N.A., Sagadeeva M.A., Sviridyuk G.A. Numerical Optimal Measurement Algorithm under Distortions Caused by Inertia, Resonances, and Sensor Degradation. Automation and Remote Control, 2021, vol. 82, no. 1, pp. 41-50. DOI: 10.1134/S0005117921010021
27. Shestakov A.L., Keller A.V. Optimal Dynamic Measurement Method Using Digital Moving Average Filter. Journal of Physics, 2021, vol. 2021, article ID: 012073, 7 p. DOI: 10.1088/1742-6596/1864/1/012073
28. Shestakov A.L.,Zamyshlyaeva A.A., Manakova N.A., Sviridyuk G.A., Keller A.V. Reconstruction of a Dynamically Distorted Signal Based on the Theory of Optimal Dynamic Measurements. Automation and Remote Control, 2021, vol. 82, no. 12, pp. 2143-2154. DOI: 10.1134/S0005117921120067
29. Skripnik V.P. Degenerate Linear Systems. Russian Mathematics, 1982, no. 3, pp. 62-67.
30. Sukacheva T.G., Matveeva O.P. The Problem of the Thermoconvection of an Incompressible Viscoelastic Kelvin-Voigt Fluid of Nonzero Order. Russian Mathematics, 2001, vol. 45, no. 11, pp. 44-51.
31. Sviridyuk G.A., Efremov A.A. Optimal Control for a Class of Degenerate Linear Equations. Doklady Akademii Nauk, 1999, vol. 364, no. 3, pp. 323-325.
32. Sviridyuk G.A., Brychev S.V. Numerical Solution of Systems of Equations of Leontief Type. Russian Mathematics, 2003, vol. 47, no. 8, pp. 44-50.
33. Sviridyuk G.A., Keller A.V. On the the Numerical Solution Convergence of Optimal Control Problems for Leontief Type System. Journal of Samara State Technical University. Series: Physical and Mathematical Sciences, 2011, no. 2 (23), pp. 24-33. (in Russian)
34. Zagrebina S.A., Konkina A.S. The Multipoint Initial-Final Value Condition for the Navier-Stokes Linear Model. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 1, pp. 132-136. DOI: 10.14529/mmp150111
35. Zamyshlyaeva A.A., Muravyev A.S. Computational Experiment for One Mathematical Model of Ion-Acoustic Waves. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 127-132. DOI: 10.14529/mmp150211
36. Zamyshlyaeva A.A., Manakova N.A., Tsyplenkova O.N. Optimal Control in Linear Sobolev Type Mathematical Models. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 1, pp. 5-27. DOI: 10.14529/mmp200101