Approximation of the Multidimensional Optimal Control Problem for the Heat Equation (Applicable to Computational Fluid Dynamics (CFD))
Abstract
This work is devoted to finding an estimate of the convergence rate of an algorithm for numerically solving the optimal control problem for the three-dimensional heat equation. An important aspect of the work is not only the establishment of convergence of solutions of a sequence of discrete problems to the solution of the original differential problem, but the determination of the order of convergence, which plays a very important role in applications. The paper uses the discretization method of the differential problem and the method of integral estimates. The reduction of a differential multidimensional mixed problem to a difference one is based on the approximation of the desired solution and its derivatives by difference expressions, for which the error of such an approximation is known. The idea of using integral estimates is typical for such problems, but in the multidimensional case significant technical difficulties arise. To estimate errors, we used multidimensional analogues of the integration formula by parts, Friedrichs and Poincare inequalities. The technique used in this work can be applied under some additional assumptions, and for nonlinear multidimensional mixed problems of parabolic type. To find a numerical solution, the variable direction method is used for the difference problem of a parabolic type equation. The resulting algorithm is implemented using program code written in the Python 3.7 programming language.
Keywords
References
Godunov, Sergei Konstantinovich, and Viktor Solomonovich Ryabenkii. “Difference Schemes: An Introduction to the Underlying Theory”. Elsevier, 1987.
Kostikov Yu. A., Pavlov V. Y., Romanenkov A. M. Optimization of vibration frequencies of the elastic plate in an ideal fluid. Space, Time and Fundamental Interactions, 2018, no. 1, pp. 82–91.
Muravey, Leonid A., Victor M. Petrov, and Alexandr M. Romanenkov. “The Problem of Damping the Transverse Oscillations on a Longitudinally Moving String.” Mordovia University Bulletin no. 4 (December 28, 2018): 472–485. doi:10.15507/0236-2910.028.201804.472-485.
Muravey, L. A., A. M. Romanenkov, and V. M. Petrov. "Optimal Control of Nonlinear Processes in Problems of Mathematical Physics." (2018).
Serovaiskii, S. Ya. "Approximation Methods in Optimal Control Problems for Nonlinear Infinite-Dimensional Systems." Mathematical notes 94, no. 3-4 (2013): 567-582.
Edalatzadeh, M. Sajjad, Dante Kalise, Kirsten A. Morris, and Kevin Sturm. "Shape Optimization of Actuators over Banach Spaces for Nonlinear Systems." arXiv preprint arXiv:2002.07172 (2020).
Edalatzadeh, Mohammad Sajjad, and Aria Alasty. “Boundary Exponential Stabilization of Non-Classical Micro/nano Beams Subjected to Nonlinear Distributed Forces.” Applied Mathematical Modelling 40, no. 3 (February 2016): 2223–2241. doi:10.1016/j.apm.2015.09.026.
Edalatzadeh, M. Sajjad, and Kirsten A. Morris. “Optimal Actuator Design for Semilinear Systems.” SIAM Journal on Control and Optimization 57, no. 4 (January 2019): 2992–3020. doi:10.1137/18m1171229.
Deligiannidis, G., S. Maurer, and M. V. Tretyakov. "Random walk algorithm for the Dirichlet problem for parabolic integro-differential equation." arXiv preprint arXiv:2001.05531 (2020).
Mukam, Jean Daniel, and Antoine Tambue. “Strong Convergence of the Linear Implicit Euler Method for the Finite Element Discretization of Semilinear Non-Autonomous SPDEs Driven by Multiplicative or Additive Noise.” Applied Numerical Mathematics 147 (January 2020): 222–253. doi:10.1016/j.apnum.2019.08.009.
Guillén-González, Francisco, Exequiel Mallea-Zepeda, and Élder J. Villamizar-Roa. "On a bi-dimensional chemo-repulsion model with nonlinear production." arXiv preprint arXiv:2002.03199 (2020).
Biccari, Umberto, Víctor Hernández-Santamaría, and Judith Vancostenoble. "Existence and cost of boundary controls for a degenerate/singular parabolic equation." arXiv preprint arXiv:2001.11403 (2020).
Vladimirov, V. S. “Equations of Mathematical Physics” (1981): 512.
Shubin, M. A. “Lectures on the Equations of Mathematical Physics” (2003): 303.
Yongbin, G., Fei, Z., and Jianying, W. “A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems.” Applied and Computational Mathematics 7, no. 1 (2018): 1. doi:10.11648/j.acm.20180701.11.
V. S. Ryaben’kiy, Introduction to Computational Mathematics (in Russian), Nauka, Moscow (2000):296.
Vasilev, F. P. "Optimization Methods: Minimization in Functional Spaces, Regularization and Approximation." (2002).
DOI: 10.28991/cej-2020-03091506
Refbacks
- There are currently no refbacks.
Copyright (c) 2020 Yuri Alexandrovich Kostikov, Alexander Mikhailovich Romanenkov
This work is licensed under a Creative Commons Attribution 4.0 International License.
