Numerical solution of the optimal speed problem with a phase constraint for one parabolic equation

Authors

DOI:

https://doi.org/10.15587/2312-8372.2017.112752

Keywords:

parabolic equation, speed problem, optimal control, penalty functional, gradient of the functional

Abstract

An algorithm for the numerical solution of the optimal speed problem with phase constraint for a parabolic equation describing the heat conduction processes in inhomogeneous media is proposed. To solve the problems with the use of first-order optimization methods and finite differences on non-uniform grids, analytical formulas are obtained for the components of the gradient of the functional with respect to controllable functions. A method is proposed for selecting initial approximations for optimal controls and a step in time at each iteration, which makes it possible to accelerate the computation process. To achieve the specified accuracy, the speed problem requires 6 iterations and . Based on the analysis of the results of numerical experiments, the influence of various parameters on the iterative process is investigated and recommendations are developed on the use of the proposed algorithm. In optimal control problems, the total number of iterations in option a by the conditional gradient method is 110 and the gradient projection method is 108. In option b, the total number of iterations is CGM – 81, GPM – 64, i. e., the total number of iterations in the optimal control problem in method b the choice of the initial approximation is much less than in variant a.  The optimal speed control, obtained by both methods, is close enough to test controls. Numerical experiments are also carried out in the case when the control-optimal controls have two switching points. However, the nature of the results obtained does not change. The proposed algorithm can be used to determine the optimal regime and time of thermal conductivity processes in inhomogeneous media.

Author Biographies

Saftar Ilyas Huseynov, Azerbaijan State University of Oil and Industry, 20, Azadlig ave., Baku, Azerbaijan, AZ1010

PhD, Associate Professor

Department of General and Applied Mathematics

Sardar Yusub Qasimov, Azerbaijan State University of Oil and Industry, 20, Azadlig ave., Baku, Azerbaijan, AZ1010

PhD, Associate Professor

Department of General and Applied Mathematics

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Published

2017-09-21

How to Cite

Huseynov, S. I., & Qasimov, S. Y. (2017). Numerical solution of the optimal speed problem with a phase constraint for one parabolic equation. Technology Audit and Production Reserves, 5(1(37), 42–47. https://doi.org/10.15587/2312-8372.2017.112752

Issue

Vol. 5 No. 1(37) (2017): Industrial and Technology Systems

Section

Technology and System of Power Supply: Original Research

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Copyright (c) 2017 Sardar Yusub Qasimov, Saftar Ilyas Huseynov

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