Research article


Oleg Sergeyevich Ablyalimov, Anna Nikolayevna Avdeyeva, Otabek Rustamovich Khamidov, Obidjan Tairdjanovich Kasimov

Online First: December 30, 2022

In this study, we discuss methods for solving optimal control problems based on neural network methods. We are studying a hierarchical dynamic two-level surface water quality control system. The system consists of a supervisory authority (government) and several agents (enterprises). We consider this problem from the point of view of agents. In this case, we solve the optimal control problem with constraints. To solve this problem, we use the Pontryagin maximum principle, by which we obtain optimality conditions. To solve the emerging ODES, we use a live broadcast neural network. We provide an overview of existing methods for studying such problems and an overview of neural network training methods. To estimate the error of the numerical solution, we propose to use a defect analysis method adapted for neural networks. This allows us to obtain quantitative estimates of the error of the numerical solution. We give examples of using our method to solve a synthetic problem and a model of surface water quality control. We compare the results of these examples with the known solution (when it is provided) and the results of the survey method. In all cases, the errors estimated by our method are of the same order as the errors compared to the known solution. Moreover, we study the problem of surface water quality control when other methods do not provide a solution. This is due to the relatively large time interval and/or the case of multiple agents. In the latter case, we are looking for a Nash equilibrium between the agents. Thus, in this study we show the ability of neural networks solve various problems, including optimal control problems and differential games, and we show the ability to quantify the error. From the numerical results, we conclude that the presence of a supervisor is necessary to achieve sustainable development.


Optimal control, differential games, neural network, Nash equilibrium, Pontryagin’s maximum principle