A numerical method for solving boundary optimal control problem modeled by heat transfer equation, in the presence of a scale invariance property

Karim, Benalia; Karim, Beddek; Brahim, Oukacha

doi:10.22124/jmm.2024.27197.2403

A numerical method for solving boundary optimal control problem modeled by heat transfer equation, in the presence of a scale invariance property

Document Type : Research Article

Authors

¹ Department of Mathematical Sciences, Faculty of Hydrocarbons and Chemistry, University of Boumerdes, Boumerdes, Algeria

² Laboratory of Applied Automation (LAA), Department of Automation, Faculty of Hydrocarbons and Chemistry, University of Boumerdes, Algeria

³ Laboratory of Operational Research and Mathematical decision, University of Tizi-Ouzou, Hasnaoua II 15000, Algeria

10.22124/jmm.2024.27197.2403

Abstract

In this paper, we present a computational approach for solving a boundary optimal control problem modeled by heat transfer equation with two-point boundary conditions, in the presence of a scale invariance property under dilation. First, we establish a scale-invariant solution. Indeed, the dependence of this solution towards a scale invariance factor naturally leads to an optimal control problem. Second, we propose a numerical approach to solve this problem. The idea consists in transforming the problem into an optimal control problem modeled by a system of ordinary differential equations invariant under dilation using the finite difference approximation. Therefor, the minimum principle of Pontryagin is applied to derive the necessary optimality conditions that are solved by the vartiational iteration method to get an approximate scale-invariant solutions for the optimal control law. Finally, to show the efficiency of this approach, a numerical example is illustrated and comparison with another method is performed.

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