{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,5]],"date-time":"2025-11-05T17:26:27Z","timestamp":1762363587604,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"CIDMA under the Portuguese Foundation for Science and Technology","award":["UID\/4106\/2025","UID\/PRR\/4106\/2025"],"award-info":[{"award-number":["UID\/4106\/2025","UID\/PRR\/4106\/2025"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper presents a novel approach to the controllability of nonlinear dynamic systems using recurrent neural networks (RNNs). We develop a comprehensive theoretical framework that integrates controllability analysis, stability verification via Lyapunov functions, and the derivation of optimal control laws based on Pontryagin\u2019s Maximum Principle. Our methodology not only ensures theoretical soundness but also offers practical effectiveness. To demonstrate its applicability, we conduct simulations using real-world data from the AAPL stock database. The proposed RNN-based control framework significantly reduces the deviation between predicted system outputs and actual observations. We further enhance performance through two complementary strategies, a direct control method and a parameter optimization approach, both of which contribute to the accuracy and adaptability of the control system. These results confirm the potential of neural network-based control in managing complex nonlinear dynamics<\/jats:p>","DOI":"10.3390\/axioms14110808","type":"journal-article","created":{"date-parts":[[2025,10,31]],"date-time":"2025-10-31T05:28:43Z","timestamp":1761888523000},"page":"808","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Stability and Controllability of Nonlinear Dynamic Systems with Neural Networks: An Application to Financial Data"],"prefix":"10.3390","volume":"14","author":[{"given":"Lamiae","family":"Seddati","sequence":"first","affiliation":[{"name":"Laboratory of Applied Sciences & Emerging Technologies (LSATE), National School of Applied Sciences of Fez, Sidi Mohamed Ben Abdellah University, Avenue My Abdallah Km 5 Route d\u2019Imouzzer, Fez BP 72, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7074-764X","authenticated-orcid":false,"given":"Touria","family":"Karite","sequence":"additional","affiliation":[{"name":"Laboratory of Applied Sciences & Emerging Technologies (LSATE), National School of Applied Sciences of Fez, Sidi Mohamed Ben Abdellah University, Avenue My Abdallah Km 5 Route d\u2019Imouzzer, Fez BP 72, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3599-9099","authenticated-orcid":false,"given":"Ahmed","family":"Aberqi","sequence":"additional","affiliation":[{"name":"Laboratory of Applied Sciences & Emerging Technologies (LSATE), National School of Applied Sciences of Fez, Sidi Mohamed Ben Abdellah University, Avenue My Abdallah Km 5 Route d\u2019Imouzzer, Fez BP 72, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nuno R. O.","family":"Bastos","sequence":"additional","affiliation":[{"name":"School of Technology and Management of Viseu, Polytechnic Institute of Viseu, Campus Polit\u00e9cnico, 3504-510 Viseu, Portugal"},{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, Campus Universit\u00e1rio de Santiago, 3810-193 Aveiro, Portugal"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/S0092-8240(05)80006-0","article-title":"A logical calculus of the ideas immanent in nervous activity","volume":"52","author":"McCulloch","year":"1990","journal-title":"Bull. Math. Biol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2557","DOI":"10.1073\/pnas.79.8.2554","article-title":"Neural networks and physical systems with emergent collective computational abilities","volume":"79","author":"Hopfield","year":"1982","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3088","DOI":"10.1073\/pnas.81.10.3088","article-title":"Neurons with graded response have collective computational properties like those of two-state neurons","volume":"81","author":"Hopfield","year":"1984","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/BF00339943","article-title":"Neural Computation of Decisions in Optimization Problems","volume":"52","author":"Hopfield","year":"1985","journal-title":"Biol. Cybern."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1257","DOI":"10.1109\/31.7600","article-title":"Cellular neural networks: Theory","volume":"35","author":"Yang","year":"1988","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1273","DOI":"10.1109\/31.7601","article-title":"Cellular neural networks: Applications","volume":"35","author":"Yang","year":"1988","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"965","DOI":"10.1016\/j.chaos.2005.02.003","article-title":"On Global Exponential Stability of Nonautonomous Delayed Neural Networks","volume":"26","author":"Zhang","year":"2005","journal-title":"Chaos Solitons Fractals"},{"key":"ref_8","first-page":"364","article-title":"Stability analysis of recurrent neural networks with bounded activation functions","volume":"332","author":"Yang","year":"2004","journal-title":"Phys. Lett. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.cam.2003.11.014","article-title":"Existence and Stability of Equilibrium of the Continuous-Time Hopfield Neural Networks","volume":"169","author":"Chen","year":"2004","journal-title":"J. Comput. Appl. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1016\/0893-6080(89)90018-X","article-title":"Convergent activation dynamics in continuous time networks","volume":"2","author":"Hirsch","year":"1989","journal-title":"Neural Netw."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/0167-2789(94)90043-4","article-title":"Stability in Asymmetric Hopfield Nets with Transmission Delays","volume":"76","author":"Gopalsamy","year":"1994","journal-title":"Phys. D Nonlinear Phenom."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1016\/0893-6080(92)90011-7","article-title":"Stability Conditions for Nonlinear Continuous Neural Networks with Asymmetric Connection Weights","volume":"5","author":"Maluoka","year":"1992","journal-title":"Neural Netw."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Kaslik, E., Braescu, L., and Balint, S. (2005, January 25\u201329). On the Controllability of the Continuous-Time Hopfield-Type Neural Networks. Proceedings of the Seventh International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC\u201905), Timisoara, Romania.","DOI":"10.1109\/SYNASC.2005.53"},{"key":"ref_14","unstructured":"Slotine, J.J.E., and Li, W. (1991). Applied Nonlinear Control, Prentice-Hall."},{"key":"ref_15","unstructured":"Khalil, H.K. (2002). Nonlinear Systems, Prentice-Hall."},{"key":"ref_16","unstructured":"Almeida, L.B. (1989). Backpropagation in non-feedforward networks. Neural Computing Architectures, North Oxford Academic."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/323533a0","article-title":"Learning representations by back-propagating errors","volume":"323","author":"Rumelhart","year":"1986","journal-title":"Nature"},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Talebi, S.P., and Mandic, D.P. (2024, January 22\u201325). On the Dynamics of Multi-agent Nonlinear Filtering and Learning. Proceedings of the 2024 IEEE 34th International Workshop on Machine Learning for Signal Processing (MLSP), London, UK.","DOI":"10.1109\/MLSP58920.2024.10734830"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"41","DOI":"10.2514\/1.3210","article-title":"Parameter Estimation in Nonlinear Systems Using Hopfield Neural Networks","volume":"42","author":"Hu","year":"2005","journal-title":"J. Aircr."},{"key":"ref_20","first-page":"1096","article-title":"A Study on Asset Pricing in Stock Market Based on Hopfield Neural Network","volume":"11","author":"Sun","year":"2025","journal-title":"Int. J. Comput. Exp. Sci. Eng. (IJCESEN)"},{"key":"ref_21","first-page":"28","article-title":"Decision System for Stock Data Forecasting Based on Hopfield Artificial Neural Network","volume":"6","author":"Paluch","year":"2016","journal-title":"Inform. Autom. Pomiary W Gospod. I Ochr. \u015arodowiska"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Nicolini, C., Gopalan, M., Staiano, J., and Lepri, B. (2024). Hopfield Networks for Asset Allocation. arXiv.","DOI":"10.1145\/3677052.3698605"},{"key":"ref_23","unstructured":"Cummins, J.S., and Berloff, N.G. (2024). A Fully Analog Pipeline for Portfolio Optimization. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"El Moutaouakil, K., Bouhanch, Z., Ahourag, A., Aberqi, A., and Karite, T. (2024). OPT-FRAC-CHN: Optimal Fractional Continuous Hopfield Network. Symmetry, 16.","DOI":"10.3390\/sym16070921"},{"key":"ref_25","unstructured":"Jai, A.E., Zerrik, E., and Ztot, K. (2008). Syst\u00e8mes Dynamiques: Analyse et Contr\u00f4le des Syst\u00e8mes Localis\u00e9s, Presses Universitaires de Perpignan."},{"key":"ref_26","unstructured":"Tr\u00e9lat, E. (2008). Contr\u00f4le Optimal: Th\u00e9orie et Applications, Vuibert."},{"key":"ref_27","unstructured":"Chen, T. (1995). Linear Systems Theory and Design, Oxford University Press."},{"key":"ref_28","unstructured":"Battiston, M., Rea, R., Cerri, G., Paolillo, M., and Alessandro, S. (2024, October 21). yfinance: Yahoo! Finance Market Data Downloader. Available online: https:\/\/pypi.org\/project\/yfinance\/."},{"key":"ref_29","first-page":"106","article-title":"Local Volatility and Hopfield Neural Network","volume":"21","year":"2022","journal-title":"China-USA Bus. Rev."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"102238","DOI":"10.1016\/j.seps.2025.102238","article-title":"Research on the collaborative mechanism of a data trading market based on a four-party evolutionary game in the context of digital intelligence","volume":"100","author":"Li","year":"2025","journal-title":"Socio-Econ. Plan. Sci."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"He, Q., Rahman, M.U., Hu, J., and Cui, J. (2025). Dynamic relationship between carbon trading system resilience and low-carbon stock market returns using time-varying Granger causality. Fractals, 2540181.","DOI":"10.1142\/S0218348X25401814"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Wang, Y., and Luo, J. (2025). Effect and Challenge of Credit Guarantee Plan in Financing of Small Enterprises. Singap. Econ. Rev.","DOI":"10.1142\/S0217590825490116"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"107787","DOI":"10.1016\/j.jfranklin.2025.107787","article-title":"Learning economic model predictive control via clustering and kernel-based Lipschitz regression","volume":"362","author":"Xiong","year":"2025","journal-title":"J. Frankl. Inst."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"751","DOI":"10.1137\/S0363012997321358","article-title":"Finite-time stability of continuous autonomous systems","volume":"38","author":"Bhat","year":"2000","journal-title":"SIAM J. Control Optim."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1016\/S0167-6911(02)00130-5","article-title":"Finite-time control for robot manipulators","volume":"46","author":"Hong","year":"2001","journal-title":"Syst. Control Lett."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2106","DOI":"10.1109\/TAC.2011.2179869","article-title":"Nonlinear feedback design for fixed-time stabilization of linear control systems","volume":"57","author":"Polyakov","year":"2012","journal-title":"IEEE Trans. Autom. 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