###### Abstract
This limitation can be mitigated by using neural network technologies capable of efficiently processing data in large amounts. However, the need for specialized software tools makes integration with modern neural network architectures difficult. Therefore, a significant direction is the development of an adaptive neural network controller, which changes the parameters that are capable of changing in real time depending on the control object dynamics [19; 20; 21; 22; 23; 24]. However, most existing approaches [19; 20] are limited to using static switching functions, which do not allow for the system's nonlinear behaviour to be fully considered. This reduces adaptation efficiency in complex transient conditions [20] or causes extensive time delays [21]. Known neural network controllers are developed based on the following neural network architectures: RBF networks [20], Jordan neural networks [21], Elman neural networks [22], Hopfield neural networks [23] and Hamming neural networks [24]. Known neural network controllers' key drawback is their limited ability to integrate sensitivity functions that are sensitive to changes in the control object's dynamics. For example, RBF networks do not adapt effectively enough to rapidly changing conditions, and Elman and Jordan neural networks require significant computing resources to account for complex nonlinear dynamics in real time. Thus, the issue of integrating sensitivity functions with modern neural network technologies has not been sufficiently overcome. Despite the success in the application of second-order sensitivity methods, the task of developing algorithms that would be researched effectively under extensive delay conditions, strong nonlinearities and high uncertainty remains relevant. In addition, most research [5; 8; 9; 10; 13; 14; 16; 19; 20] is aimed at analyzing stable control objects, which limits the application of existing methods to systems with changing characteristics. Another unsolved problem is the selection and optimization of the controller switching function. The influence of different function options on the quality of transient processes has not been researched sufficiently [6; 7; 12; 15; 17], which limits the controllers' use in dynamically changing conditions. The search-free method for a controller's parametric adaptation described in [25; 26; 27] has a number of advantages, including the use of controller adaptive parameters, which improves the system's tuning accuracy and takes into account the interactions between the parameters. Regularization and gradient procedures with the inclusion of a dynamic training rate make this method especially useful for systems with variable structures. The main disadvantage of the search-free method [25; 26; 27; 28] is its limited ability to take into account strong nonlinearities and a control object's extensive time delay. The high computational complexity associated with the analytical calculation of second-order sensitivity functions complicates its application to systems with a variable structure [5; 6] and dynamically changing characteristics [7]. Table 1 provides a comparison of the approaches considered. To improve the search-free method [25; 26; 27; 28], it is necessary to integrate neural network approaches for automated modelling and the numerical computation of second-order sensitivity functions, which will reduce the computational complexity and increase accuracy. It is vital to take into account the nonlinearities and the control object's delays by implementing adaptive switching functions in the controller operator. It is also necessary to automate the adaptation of controller parameters using error backpropagation methods modified for dynamic systems and implement a dynamic training rate mechanism to accelerate convergence. To improve the optimization stability, regularization should be included in the computation of the Hessian matrix and the optimality criterion, which will prevent overtraining and enhance system reliability in uncertain and noisy environments.