DOI 10.15507/2079-6900.24.202202.162-174

Original article

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

MSC2020 93-08

On an iterative method for solving optimal control problems for an elliptic type system

M. E. Fairuzov, F. V. Lubyshev

Bashkir State University (Ufa, Russian Federation)

Abstract. An important class of applied problems is that of optimal control of some objects’ state. It is required to select control actions in such a way as to achieve desired effect. We deal with distributed systems, since their state is described by a partial differential equation. In this paper we study an iterative process for solving the problem of optimal control for an elliptic type system. Similar problem arises during the control of thermal processes. The quality of system state control is estimated by a given quality functional defined on the solution of the Dirichlet problem for an elliptic equation. One of the most important classes of thermal process control problems is temperature control, which means maintaining given temperature in the computational domain due to certain thermal effects. Here, a distributed internal heat source acts as a control. In the paper, we study statement correctness of the optimal control problem with a regularized functional. More precisely, we examine control problem for a system described by an elliptic type equation and formulate its optimality condition in the form of a system of equations for initial and conjugate states. An iterative method is proposed for solving the optimal control problem of an elliptic type system. Convergence of the iterative process is studied, and the rate of convergence is estimated.

Key Words: optimal control, elliptic equations, boundary value problem, iterative method, convergence of the iterative method

For citation: M. E. Fairuzov, F. V. Lubyshev. On an iterative method for solving optimal control problems for an elliptic type system. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 24:2(2022), 162–174. DOI: https://doi.org/10.15507/2079-6900.24.202202.162-174

Submitted: 22.03.2021; Revised: 27.04.2022; Accepted: 25.05.2022

Information about the authors:

Mahmut E. Fairuzov, Associate Professor, Department of Information Technology and Computer Mathematics, Bashkir State University (32 Zaki Validi St., Ufa 450076, Russia), Ph.D. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-9118-660X, fairuzovme@mail.ru

Fedor V. Lubyshev, Professor, Department of Information Technology and Computer Mathematics, Bashkir State University (32 Zaki Validi St., Ufa 450076, Russia), Dr.Sci. (Phys.-Math.), ORCID: http://orcid.org/0000-0002-3279-4293, maxam721@mail.ru

All authors have read and approved the final manuscript.

Conflict of interest: The authors declare no conflict of interest.

Creative Commons Attribution 4.0 International License This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License.