Document Type : Research Article
Author
Department of Applied Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran.
10.30473/coam.2025.74474.1306
Abstract
This paper presents a hybrid scheme for solving optimal control problems. Discretizing the time interval and assuming a constant control value on each sub-interval transforms the optimal control problem into an assignment problem. To cluster feasible solutions, a novel method is proposed in this paper, which applies metaheuristic algorithms—specifically, genetic algorithms and particle swarm optimization—to generate a large number of solutions. Subsequently, the $K$-means clustering method is employed to classify these solutions into clusters. Enhancing the median of each cluster, using metaheuristic techniques, ultimately results in improved medians. The best median from the final iteration of the algorithm serves as an acceptable solution for the optimal control problem. In some cases, it even succeeds in discovering a new best solution.
Highlights
- A hybrid optimization framework is introduced for optimal control that integrates time discretization, clustering, and metaheuristics.
- A two-stage approach uses genetic algorithms (GA) and particle swarm optimization (PSO) to generate diverse feasible solution populations for the control problem.
- The solution quality is enhanced by applying GA crossover and PSO directional updates to refine candidates within and across clusters.
- The optimal control problem is reformulated as selecting the best representative solution from clustered populations via a K-means-based P-median mechanism, improving tractability and performance.
- The method is validated on multiple optimal control problems, exploring three cluster counts and two time-interval discretizations, with several instances achieving superior results compared to existing literature.
Keywords
- Clustering
- K-means algorithm
- Optimal control problem
- Genetic algorithm
- Particle swarm optimization
Main Subjects
[1] Alfiyatin, A.N., Mahmudy, W.F., Anggodo, Y.P. (2018). “K-Means clustering and genetic algorithm to solve vehicle routing problem with time windows”, Indonesian Journal of Electrical Engineering and Computer Science, 11(2), 462-468, doi:http://doi.org/10.11591/ijeecs.v11.i2.pp462-468.
[2] Borzabadi, A.H., Mehne, H.H. (2009). “Ant colony optimization for optimal control problems”, Journal of Information and Computing Science, 4(4), 259-264, doi:https://doi.org/2024-JICS-22734.
[3] Borzabadi, A.H., Mirassadi, S., Heidari, M. (2014). “Population based algorithms for approximate optimal distributed control of wave equations“, Iranian Journal of Numerical Analysis and Optimization, 4(2), 31-41, doi:https://doi.org/10.22067/ijnao.v4i2.40230.
[4] Borzabadi, A.H., Ghasemi, S.G., Fard, O.S. (2011). “A hybrid iterative scheme for optimal control problems based on L-M and penalty techniques”, Journal of Advanced Research in Dynamical and Control Systems, 3(2), 53-65.
[5] Cattrysse, D.G., Van Wassenhove, L.N. (1992). “A survey of algorithms for the generalized assignment problem”, European Journal of Operational Research, 60(3), 260-272, doi:https://doi.org/10.1016/0377-2217(92)90077-m.
[6] Diveev, A., Sofronova, E., Konstantinov, S. (2021). “Approaches to numerical solution of optimal control problem using evolutionary computations”, Applied Sciences, 11(15), 7096, doi:https://doi.org/10.3390/app11157096.
[7] Farahani, M.A., McKendall, A. (2023). “A Genetic Algorithm Meta-Heuristic for a Generalized Quadratic Assignment Problem”, Preprint available at arXiv:2308.07828, doi:http://doi.org/10.48550/arXiv.2308.07828.
[8] Fard, O.S., Borzabadi A.H. (2007). “Optimal control problem, quasi-assignment problem and genetic algorithm”, Enformatika, Transaction on Engineering Computations and Technologies, 19, 422-424.
[9] Gaon, T., Gabay, Y., Cohen, M.W. (2025). “Optimizing electric vehicle routing efficiency using K-means clustering and genetic algorithms”, Future Internet, 17(3), 97, doi:https://doi.org/10.3390/fi17030097.
[10] Greistorfer, P., Staněk, R., Maniezzo, V. (2022). “A tabu search matheuristic for the generalized quadratic assignment problem”, In: Di Gaspero, L., Festa, P., Nakib, A., Pavone, M. (eds) Metaheuristics. MIC 2022. Lecture Notes in Computer Science, vol 13838. Springer, Cham, doi:https://doi.org/10.1007/978-3-031-26504-4_46.
[11] Hatamian, R., Samareh Hashemi, S.A. (2025). “A hybrid numerical approach for solving nonlinear optimal control problems”, Control and Optimization in Applied Mathematics, 10(1), 125-138, doi: https://doi.org/10.30473/coam.2025.72875.1273.
[12] Jain, A.K. (2010). “Data clustering: 50 years beyond K-means”, Pattern Recognition Letters, 31(8), 651-666, doi:https://doi.org/10.1016/j.patrec.2009.09.011.
[13] Konstantinov, S., Diveev, A. (2021). “Evolutionary algorithms for optimal control problem of mobile robots group interaction”, In: Olenev, N.N., Evtushenko, Y.G., Jaćimović, M., Khachay, M., Malkova, V. (eds) Advances in Optimization and Applications. OPTIMA 2021. Communications in Computer and Information Science, vol 1514. Springer, Cham, doi:https://doi.org/10.1007/978-3-030-92711-0_9.
[14] Lucrezia, M. (2024). “Application of optimal control techniques to the parafoil flight of space rider”, Aerotecnica Missili & Spazio, 103(2), 117-128, doi:https://doi.org/10.1007/s42496-023-00176-3.
[15] Massaro, M., Lovato, S., Bottin, M., Rosati, G. (2023). “An optimal control approach to the minimum-time trajectory planning of robotic manipulators”, Robotics, 12(3), 64, doi:https://doi.org/10.3390/robotics12030064.
[16] Mehne, H.H., Borzabadi, A.H. (2006). “A numerical method for solving optimal control problems using state parametrization”, Numerical Algorithms, 42, 165-169, doi:https://doi.org/10.1007/s11075-006-9035-5.
[17] Mînzu, V., Arama, I. (2022). “Optimal control systems using evolutionary algorithm-control input range estimation”, Automation, 3(1), 95-115, doi:https://doi.org/10.3390/automation3010005.
[18] Salimi, M., Borzabadi, A.H., Mehne, H.H., Heydari, A. (2021). “The hub location’s method for solving optimal control problems”, Evolutionary Intelligence, 14, 1671-1690, doi:https://doi.org/10.1007/s12065-020-00437-1.
[19] Salimi, M., Borzabadi, A.H., Mehne, H.H., Heydari, A. (2025). “A parallel hybrid variable neighborhood descent algorithm for non-linear optimal control problems”, Iranian Journal of Numerical Analysis & Optimization, 15(2), doi:https://doi.org/10.22067/ijnao.2024.86859.1389.
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