The small parameter method in the optimisation of a quasi-linear dynamical system problem

  • Anatoly I. Kalinin Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
  • Leonid I. Lavrinovich Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus https://orcid.org/0000-0002-7698-0207
  • Darya Y. Prudnikova Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
DOI: https://doi.org/10.33581/2520-6508-2022-2-23-33

Abstract

We consider an optimisation problem for the transient process in a quasi-linear dynamical system (contains a small parameter at non-linearities) with a performance index that is a linear combination of energy costs and the duration of the process. An algorithm for constructing asymptotic approximations of a given order to the solution of this problem is proposed. The algorithm is based on the asymptotic decomposition by integer powers of a small parameter of the initial values of adjoint variables and the duration of the process that are finite-dimensional elements, according to which the solution of the problem is easily restored. The computational procedure of the algorithm includes solving the problem of optimising the transient process in a linear dynamical system, integrating systems of linear differential equations, and finding the roots of non-degenerate linear algebraic systems. We also show how the constructed asymptotic approximations can be used to construct optimal control in the problem under consideration for a given value of a small parameter.

Author Biographies

Anatoly I. Kalinin, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

doctor of science (physics and mathematics), full professor; professor at the department of optimal control methods, faculty of applied mathematics and computer science

Leonid I. Lavrinovich, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

PhD (physics and mathematics), docent; associate professor at the department of optimal control methods, faculty of applied mathematics and computer science

Darya Y. Prudnikova, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

student at the faculty of applied mathematics and computer science

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Published
2022-06-24
Keywords: small parameter, quasi-linear system, optimal control, asymptotic approximations
How to Cite
Kalinin, A. I., Lavrinovich, L. I., & Prudnikova, D. Y. (2022). The small parameter method in the optimisation of a quasi-linear dynamical system problem. Journal of the Belarusian State University. Mathematics and Informatics, 2, 23-33. https://doi.org/10.33581/2520-6508-2022-2-23-33
Issue
No 2 (2022): Journal of the Belarusian State University. Mathematics and Informatics
Section
Differential Equations and Optimal Control

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