NUMERICAL APPROXIMATION OF OPTIMAL CONTROL PROBLEMS CONSTRAINED BY DYNAMIC EQUATIONS VIA GALERKIN METHOD
Keywords:
Unconstrained problem, orthogonal vector, Hamiltonian, System of equations, Galerkin method, Tolerance
Abstract
The research investigates the application of the Galerkin method to optimal control problems constrained by coupled dynamic equations. These constrained problems are reformulated into unconstrained ones using the Hamiltonian approach, which facilitates the determination of boundary conditions for both the state and costate variables. By assuming a polynomial solution, the weighted and residual functions were derived. The Orthogonality of the product of these functions leads to the formation of a system of linear equations. Solving these equations provides the solution for the boundary conditions through direct substitution. This scheme was developed for the Lagrange form of optimal control problems to assess its accuracy in approximating exact solutions. Several optimal control problems with known exact solutions were solved using the proposed scheme, and the results were compared to evaluate its effectiveness.
References
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[3] M. F. Akinmuyise Embedded Multiplier in Newton’s Method For the Solution of Time Vari- ant Optimal Control Problems, faculty of science annual Conference, Ondo State Univer- sity of Science and Technology, on sustainable development through scientific innovation: Problems and Prospect, Okitipupa, Nigeria, 16th-18th July, 2019.
[4] M. F. Akinmuyise, S. O. Olagunju, S. A. Olorunsola Numerical Solution of Optimal Control Problems Using Improved Euler Method Abacus Mathematics science series, 2(2022), 338– 349.
[5] M. F. Akinmuyise, O. B.. Longe First Order Differential Equations with Applications to Security Service Companies, Journal of Advances in Mathematical and Computational Sciences, 11(2023), 1-10.
[6] M. F. Akinmuyise, K. J.,Adebayo, I. O. Adisa, E. T. Olaosebikan, S. O. Gbenro, A.
O. Dele-Rotimi, A. A. Obayomi, S. O. Ayinde Extended Runge-Kutta Method (ERKM) Algorithm for the Solution of Optimal Control Problems (OCP), Twist journal, 19( 2024), 471-478.
[7] M Dokuchaev, G. Zhou, S. Wary, S Modification of Galerkin’s method for Optimum pricing Journal of industrial and Management Optimization, 18(2021).
[8] J. S. Arora Introduction to Optimum Design, Elsevier Publication, U.S.A, 2011.
[9] D. N. Burghes, A. Graham Introduction to Control Theory including Optimal Control, 3rd Edition, Ellis Wiley and son, New York, USA, 1980, 154–196.
[10] G. L. David Optimal Control Theory for Applications, Mechanical Engineering Series,
Springer-verlag, Inc., 175 fifth Avenue, New York, 2003, 94–106
[11] E. Deeba, S. A Khuri, S. Xie An Algorithm for Solving Boundary Value Problems, Jounal of Computation in Nonlinear Science and Numerical Simulation, 16(2011), 38–49.
[12] R Fletcher Practical Methods of Optimization, 2nd edition, Wiley, New York, 2000, 229– 258.
[13] M. A. Ibiejugba, F. Otunta, S. A. Olorunsola, The role of multiplier in the then multiplier method, journal of the mathematical Society, 11(1992), 13-20.
[14] K. Issa, J. Biazar, T. O. Agboola T. Aliu Perturbed Galerkin Method for Solving Integro- Differential Equations, Journal of Applied Mathematics, 1(2022), 1–8.
[15] A. A. Kalecki Macrodynamic Theory of Business Cycles. Econometrica, 3(1935), 327-344.
[16] E. D. Kirk Optimal Control Theory, An Introduction, Dover Publication, Mineola, New York, 2004, 330-373.
[17] J. O. Olademo, A. A. Ganiyu, M. F. Akinmuyise Fireman Numerical Solution of some Optimal Control Problems. Open access Library Journal, 2(2015), 1–14.
[18] F. A. Oliveira., Colocation and Residual Correction. Numerische Mathematik, 36(1980), 27–31.
[19] P. K. Pandey Solution of Two-point Boundary Value Problems. A numerical approach: parametric difference method, Applied Mathematics and Nonlinear Sciences 3(2018), 49– 58.
[20] J. N. Reddy An Introduction to Finite Element Method, 3rd edition, McGraw-Hill, New York, United State, 2011.
[21] U. Roman, M. S. Islam. Galerkin weighted Residual Method for Solving Fourth Order Frac- tional Differential and Integral Boundary Value Problems. Journal of Applied Mathematics and Computatio, 62022), 410–422.
[22] G. R. Rose Numerical Methods for Solving Optimal Control Problems. Masters Thesis, University of Tennessee, USA, 2015.
[23] Storn, R., On the Usage of Differential Evolution for Function Optimization. Proceeding of North American Fuzzy Information Processing, CA, USA,19 June, 1996. 519–526.
[24] G. D. Ray. Optimal control theory, Lecture Note, Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, 2014.
[25] S. S. Tripathy, K. S. Narendra. Constrained Optimization problems using Multiplier meth- ods, Journal of Optimization theory and applications, 9( 1972), 59-70.
[26] D. Zwillinger Handbook of Differential Equations 3rd edition, Academic Press, Cambridge, USA, 1997.
Published
2025-12-23
How to Cite
AKINMUYISE, M. F., SHARIMAKIN, A., & FAKUNLE, I. (2025). NUMERICAL APPROXIMATION OF OPTIMAL CONTROL PROBLEMS CONSTRAINED BY DYNAMIC EQUATIONS VIA GALERKIN METHOD. Unilag Journal of Mathematics and Applications, 4(2), 69 - 82. Retrieved from https://lagjma.unilag.edu.ng/article/view/2751
Section
Articles
Copyright (c) 2024 MATTHEW FOLORUNSHO AKINMUYISE, AKINWUMI SHARIMAKIN, ILESANMI FAKUNLE
This work is licensed under a Creative Commons Attribution 4.0 International License.
