## 7 Conclusions  
In this research work, a novel Dynamic Neural Control System (DNCS) was developed to solve the trajectory tracking problem for an autonomous underwater vehicle with 4 DOF in the presence of unmodeled dynamics, parametric uncertainties, and unknown external disturbances. A proof based on a proposed Lyapunov function candidate to demonstrate the closed-loop stability was derived. The stability formal result establishes sufficient conditions in order to guarantee the ultimate boundedness of the identification and tracking errors. These conditions involve the solution of a set of two AREs, a specific control law structure, and certain adaptive laws for the synaptic weights and for the estimate of the unknown input parameter. Besides solving the trajectory tracking problem, the proposed DNCS lets us estimate and compensate (a) unmodeled dynamics and external disturbances and (b) uncertain constant gains that multiply the control signal. The main strength of the proposed control strategy consists of solving the trajectory tracking problem with reduced knowledge of the mathematical model of the AUV (basically a reduced structure) but without any knowledge of the system parameters. A set of simulations were conducted in order to demonstrate the effectiveness and applicability of the proposed methodology. Moreover, a comparison analysis via two performance indexes of the DNCS with respect to a feedback controller and a self-tuned PD controller both without a dynamic neural compensation and a parameter estimation were also conducted. These comparisons show that the proposed DNCS technique is more effective in dealing with unmodeled dynamics, parameter uncertainties, and external disturbances. The identification capacity of the DNCS to approximate the highly nonlinear behavior of the AUV is validated in the different scenarios presented in the numerical experiments. Furthermore, the simulations also showed that the tracking and identification errors are ultimately bounded and remain in a compact residual set around zero. Furthermore, two additional performance indexes were used to measure the performance of the identification process of the AUV system through the standard deviation and RMSE criterions, reaching the conclusion that the DNCS identifier possesses great properties dealing with modeling a and parametric uncertainties. Future research directions consider the implementation of the control strategy developed in at least two underwater vehicle platforms with different characteristics to verify the performance in real-time and under unknown environments. Moreover, if we consider the more general case where the added mass terms are wave frequency-dependent as considered in hydrodynamic theory, the assumption in the mathematical model that the parameter \(b\) remains constant is unsatisfied. Hence, the DNCS developed can be extended to estimate time-varying parameters, adding for example another dynamic neural network for this purpose.  
Conceptualization, F.M. and J.S.C.-R.; data curation, J.M.V.; supervision, O.S.-H.; resources, S.S. and H.R. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Mexican Secretariat of Public Education (SEP), Mexican National Council for Science and Technology Project \"Laboratorio Nacional en Vehiculos Autonomos y Exoesqueletos 314879\" and by the Secretariat of Public Education-Center for Research and Advanced Studies of the National Polytechnic Institute Project \"Desarrollo de vehiculos aereos y submarinos para aplicaciones de inspeccion en plantas hidroelectricas utilizando navegacion relativa 241\".  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.