Abstract:
The article contains a survey of publications studying problems characterized by the presence of fast variables with various rates of change (time scales). We consider the passage to the limit from the solution of a perturbed problem to the solution of a degenerate one, asymptotic solutions of initial and boundary value problems, stability and controllability, asymptotic solutions of optimal control problems, and problems with “hidden” multi-tempo variables. In addition, problems with control constraints, game problems, and stochastic systems are given. The last section presents practical problems with multi-tempo fast motions.
Keywords:singular perturbation, multi-tempo fast variables, asymptotic expansion, control problem.
Citation:
G. A. Kurina, M. A. Kalashnikova, “Singularly perturbed problems with multi-tempo fast variables”, Avtomat. i Telemekh., 2022, no. 11, 3–61; Autom. Remote Control, 83:11 (2022), 1679–1723
\Bibitem{KurKal22}
\by G.~A.~Kurina, M.~A.~Kalashnikova
\paper Singularly perturbed problems with multi-tempo fast variables
\jour Avtomat. i Telemekh.
\yr 2022
\issue 11
\pages 3--61
\mathnet{http://mi.mathnet.ru/at15860}
\crossref{https://doi.org/10.31857/S0005231022110010}
\edn{https://elibrary.ru/KDWMLX}
\transl
\jour Autom. Remote Control
\yr 2022
\vol 83
\issue 11
\pages 1679--1723
\crossref{https://doi.org/10.1134/S00051179220110017}
Linking options:
https://www.mathnet.ru/eng/at15860
https://www.mathnet.ru/eng/at/y2022/i11/p3
This publication is cited in the following 12 articles:
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A. I. Kalinin, L. I. Lavrinovich, “Asymptotic method for solving the problem of transition process optimization in a three-tempo singularly perturbed system”, Dokl. Akad. nauk, 68:3 (2024), 183
O. B. Tsekhan, “Composite Observer of a Linear Time-Varying Singularly Perturbed System with Quasidifferentiable Coefficients”, Autom Remote Control, 85:4 (2024), 341
Galina Kurina, Springer Proceedings in Physics, 315, Proceedings of the 2nd International Conference on Nonlinear Dynamics and Applications (ICNDA 2024), Volume 2, 2024, 220
Galina Kurina, 2024 10th International Conference on Control, Decision and Information Technologies (CoDIT), 2024, 2536
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G. A. Kurina, N. T. Hoai, “Zero-Order Asymptotics for the Solution of One Type of Singularly Perturbed Linear–Quadratic Control Problems in the Critical Case”, Proc. Steklov Inst. Math. (Suppl.), 321, suppl. 1 (2023), S154–S169
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A. R. Danilin, O. O. Kovrizhnykh, “Asymptotics of a Solution to an Optimal Control Problem with a Terminal Convex Performance Index and a Perturbation of the Initial Data”, Proc. Steklov Inst. Math. (Suppl.), 323, suppl. 1 (2023), S85–S97
V. A. Sobolev, “Decomposition of singularly perturbed optimal tracking problems with a given reference trajectory”, J. Appl. Industr. Math., 17:3 (2023), 640–650