## 1 Introduction
Related to this matter, References [21; 22; 23] make use of neural networks (NNs) and fuzzy inference systems (FISs) as well as mixed structures of these approaches. NNs represent a suitable approach to control complex nonlinear systems due to their capacity to deal with problems characterized by an incomplete model information or when the system to be controlled is completely unknown (a black box). Multiple controller structures based on neural networks have been postulated and deployed in previous decades for the AUV control problem [24; 25]. In the related literature, it is observed that the adaptive techniques based on neural networks have been widely used for controlling AUVs. In [26], a fault-tolerant trajectory tracking control based on dynamic programming for an AUV subject to rudders faults and disturbances produced by ocean currents is addressed. In the proposed solution, the performance index function for the optimal control problem is constructed by using estimates of faults and disturbances provided by two dynamic neural network estimators. In [27], an adaptive control system based on S surface recurrent neural networks with wavelet activation functions is designed. This control law aims to approximate the uncertain AUV dynamics introduced by a tether and a robotic manipulator. The study reported in [28] deals with the tracking control problem in a three-dimensional space of an underactuated AUV subject to external disturbances and parametric uncertainties. The proposed solution makes use of a command-filtered backstepping control scheme and an adaptive neural network control technique. In this case, the neural network is used to estimate the parametric system uncertainties in order to cancel their effect in the closed-loop system. The work in [29] presents a backstepping control based on Lyapunov theory for a Trans-media Aerial Underwater Vehicle (TMAUV), which evolves in a two dimensional space. A neural network-based adaptive estimation of model, parameter uncertainties, and external disturbances regarding the coupled input condition of the vehicle is proposed. Additionally, in [30], an adaptive tracking controller based on a combination of backstepping and terminal sliding mode techniques for a tethered AUV is designed. This type of vehicle suffers from unknown underwater disturbances as well as time-variant nonlinear cable dynamics. A recurrent neural network with radial basis functions and a disturbance observer provides uncertainty estimates. In the same way, several studies related to the design of controllers, observers, and identifiers based on a special type of dynamic neural networks have been presented in [31; 32; 33; 34]. The approach used in these works implements a type of adaptive identifier based on Hopfield-type dynamic neural networks structure introduced in [35], also called _Differential Neural Networks_ (DNNs). This name is due to the fact that the considered NNs and the dynamical control systems (subject to modeling and/or parametric uncertainties) evolve over continuous time. One of the advantages of Hopfield neural networks is that they have a simple structure, which facilitates the selection of the number of neurons required to identify a given system model, that is, the number of the system states to be identified corresponds to the number of network neurons. The static (feedforward) neural networks [36; 37; 38] have many drawbacks; for example, the function approximation is sensitive to the training data. In contrast, the dynamic (recurrent) neural networks can overcome this disadvantage because their structure incorporates feedback. Moreover, the Hopfield model describes neural networks that evolves in continuous time, i.e., this type of network can be represented as a set of continuous states, which are obtained as a solution of a set of ordinary differential equations (or difference equations). Therefore, dynamical neural networks exhibit great properties that have been exploited in the areas of identification and control of highly nonlinear dynamic systems [39; 40].