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 ... P. 127-142

We consider a linear–quadratic control problem in which there is the second power of a small parameter at the derivative of the state variable and the first power of the parameter both in the control term of the state equation and at the quadratic form of the control variable in the performance index; moreover, the state equation represents a critical case of singular perturbation theory. A zero-order asymptotic expansion of the solution is constructed using the so-called direct scheme method, in which a postulated asymptotic expansion of the solution is substituted directly into the problem statement and problems for finding the asymptotic terms are stated.

Keywords: linear–quadratic control problem, singular perturbations, critical case, asymptotics of solution

Received January 25, 2023

Revised February 15, 2023

Accepted February 20, 2023

Funding Agency: The research of the first author was supported by the Russian Science Foundation (project no. 21-11-00202), and the research of the second author was supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (project no. 101.02-2021.43).

Galina Alekseevna Kurina, Dr. Phys.-Math. Sci., Prof., Voronezh State University, Voronezh, 394018 Russia; Federal Research Center “Informatics and Control” , Russian Academy of Sciences, Moscow, 119333 Russia, e-mail: kurina@math.vsu.ru

Nguyen Thi Hoai, PhD. Phys.-Math., VNU, University of Science, Hanoi, Vietnam, e-mail: nguyenthihoai@hus.edu.vn

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Cite this article as: 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, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 127–142; Proceedings of the Steklov Institute of Mathematics  (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S154–S169.