Exploration of Physics Informed Neural Network for Solving Optimal Tracking Control Problems

[1] Ahmadin, A., Naiborhu, J., Mu’tamar, K. (2023). “Control design on a non-minimum phase bilinear system by output redefinition and particle swarm optimization method”. In: 2023 6th International Conference on Robotics, Control and Automation Engineering (RCAE), Suzhou, China IEEE., 201–206. DOI: https://doi.org/10.1109/RCAE59706.2023.10398810

[2] Ali, I. (2025). “Data-driven machine learning approach based on physics-informed neural network for population balance model”. Advances in Continuous and Discrete Models, 2025(12). DOI: https://doi.org/10.1186/s13662-025-03876-1

[3] Beard, R.W., Saridis, G.N., Wen, J.T. (1997). “Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation”. Automatica, 33(12), 2159–2177. DOI: https://doi.org/ 10.1016/S0005-1098(97)00128-3

[4] Bryson, A. E., Ho, Y. C. (1975). Applied optimal control, Taylor and Francis Group.

[5] Chen, X., Wang, F. (2021). “Neural-network-based stochastic linear quadratic optimal tracking control scheme for unknown discrete-time systems using adaptive dynamic programming”. Control Theory and Technology, 19(3), 315-327. DOI: https://doi.org/10.1007/ s11768-021-00046-y

[6] Chu, H., Miyatake, Y., Cui, W., Wei, S., Furihata, D. (2024). “Structure-preserving physicsinformed neural networks with energy or Lyapunov structure”. Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence (IJCAI-24), 3872-3880. DOI: https: //doi.org/10.24963/ijcai.2024/428

[7] Cong, A., Jin, Y., Lu, Z., Gao, Q., Ge, X., Li, Z., Lin, R., Hu, X., Hou, L. (2025). “Transfer learning-based physics-informed DeepONets for the adaptive evolution of digital twin models for dynamic systems”. Nonlinear Dynamics, 113(15), 19075–19102. DOI: https://doi.org/10. 1007/s11071-025-11158-4

[8] Devasia, S., Chen, D., Paden, B. (1996). “Nonlinear inversion-based output tracking”. IEEE Transactions on Automatic Control, 41(7), 930-942. DOI: https://doi.org/10.1109/9.508898

[9] Fabiani, G., Bollt, E., Siettos, C., Yannacopoulos, A.N. (2025). “Linear stability analysis of physics-informed random projection neural networks for ODEs”, arXiv. DOI: https://doi.org/ 10.48550/arXiv.2408.15393

[10] Feng, Y., Eun, J., Kim, S., Kim, Y.R. (2025). “Application of physics-informed neural networks (PINNs) solution to coupled thermal and hydraulic processes in silty sands”. International Journal of Geo-Engineering, 16(3). DOI: https://doi.org/10.1186/s40703-025-00232-w

[11] Gao, B., Yao, R., Li, Y. (2025). “Physics-informed neural networks with adaptive loss weighting algorithm for solving partial differential equations”. Computers & Mathematics with Applications, 181, 216-227. DOI: https://doi.org/10.1016/j.camwa.2025.01.007

[12] Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L. (2021). “Physicsinformed machine learning”. Nature Reviews Physics, 3, 422–440. DOI: https://doi.org/10. 1038/s42254-021-00314-5

[13] Lewis, F.L., Vrabie, D.L., Syrmos, V.L. (2012). “Optimal control”. 3rd Edition, John Wiley & Sons, INC.

[14] Li, Y., Liu, L. (2024). “Physics-informed neural network-based nonlinear model predictive control for automated guided vehicle trajectory tracking”. World Electric Vehicle Journal, 15(10), 460. DOI: https://doi.org/10.3390/wevj15100460

[15] Michałowska, K., Goswami, S., Karniadakis, G.E., Riemer-Sørensen, S. (2025). “Multi-resolution learning with DeepONets and long short-term memory neural networks”. Neurocomputing, 653, 131154. DOI: https://doi.org/10.1016/j.neucom.2025.131154

[16] Mu, C., Ni, Zh., Sun, C., He, H. (2017). “Air-breathing hypersonic vehicle tracking control based on adaptive dynamic programming”. IEEE Transactions on Neural Networks and Learning Systems, 28(3), 584-598. DOI: https://doi.org/10.1109/TNNLS.2016.2516948

[17] Mu’tamar, K., Naiborhu, J., Saragih, R., Handayani, D. (2022). “Tracking control for planar nonminimum-phase bilinear control system with disturbance using backstepping”. Indonesian Journal of Electrical Engineering and Computer Science, 26(3), 1315–1327. DOI: https://doi. org/10.11591/ijeecs.v26.i3.pp1315-1327

[18] Pratama, M. H. Y., Gunawan, A. Y. (2023). “Exploring physics-informed neural networks for solving boundary layer problems”. Journal of Fundamental Mathematics and Applications, 6(2), 101-116. DOI: https://doi.org/10.14710/jfma.v6i2.20084

[19] Tang, G., Fan, M. (2008). “Series-based approximate approach of optimal tracking control for nonlinear systems with time-delay”. Progress in Natural Science, 18. DOI: https://doi.org/10.1016/j.pnsc.2008.03.033

[20] Wang, D., Liu, D., Wei, Q. (2012). “Finite-horizon neuro-optimal tracking control for a class of discrete-time nonlinear systems using adaptive dynamic programming approach”. Neurocomputing, 78(1), 14-22. DOI: https://doi.org/10.1016/j.neucom.2011.03.058

[21] Wei, Q., Liu, D. (2014). “Adaptive dynamic programming for optimal tracking control of unknown nonlinear systems with application to coal gasification”. IEEE Transactions on Automation Science and Engineering, 11, 1020-1036. DOI: https://doi.org/10.1109/TASE.2013.2284545

[22] Zerrougui, I., Li, Z., Hissel, D. (2025). “Physics-informed neural network for modeling and predicting temperature fluctuations in proton exchange membrane electrolysis”. Energy and AI, 20, 100474. DOI: https://doi.org/10.1016/j.egyai.2025.100474

[23] Zhu, P., Liu, Z., Xu, Z., Lv, J. (2025). “An adaptive weight physics-informed neural network for vortex-induced vibration problems”. Buildings, 15(9), 1533. DOI: https://doi.org/10.3390/ buildings15091533