On stable solutions of the dynamical reconstruction and tracking control problems for a coupled ordinary differential equation–heat equation

Vyacheslav Maksimov

doi:10.3934/mcrf.2023005

Article Contents

2024, Volume 14, Issue 1: 322-345. Doi: 10.3934/mcrf.2023005

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On stable solutions of the dynamical reconstruction and tracking control problems for a coupled ordinary differential equation–heat equation

Vyacheslav Maksimov^,

Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia

^* Corresponding author: Vyacheslav I. Maksimov. Fax +7-343-3742581
^* Corresponding author: Vyacheslav I. Maksimov. Fax +7-343-3742581

Received: May 2022

Revised: January 2023

Early access: February 2023

Published: March 2024

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Abstract

A dynamical reconstruction problem for a system described by a coupled ordinary differential equation—heat equation is considered. The problem consists in reconstructing an unknown varying in time right-hand part of this system on the basis of inaccurate measurements of its solution. A dynamical algorithm for solving this problem is designed. An estimate for convergence rate is presented. The algorithm is stable with respect to informational noises and computational errors. It is based on the combination of the feedback control method and the method of smoothing functional well-known in the theory of ill-posed problems. The algorithm suggested in the paper is applied to solve tracking control problems. Results of a numerical experiment are discussed.

Keywords:

Dynamical inverse problem,
ODE-PDE system.

Mathematics Subject Classification: Primary: 35R30, 35Q93; Secondary: 93B30.

Citation:

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Figure 1. Scheme of control reconstruction method

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Figure 2. h = 0.001

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Figure 3. h = 0.01

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Figure 4. h = 0.001

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Figure 5. h = 0.01

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Figure 6. h = 0.001

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Figure 7. h = 0.01

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