Synthesis of Neural Network Program Control Based on Pontryagin's Maximum Principle

This paper solves the problem of synthesizing optimal control for complex nonlinear dynamic objects. Such problems often arise in various engineering applications such as robotics, mechatronics, and aerospace. Examples of such problems include the computation of the optimal flight path of an aircraft, manipulator control, and control of industrial applications. In most real-world applications, the control object is a high-order nonlinear dynamical system, and the parameters may be uncertain or vary over time.Analytical and iterative methods of optimal control synthesis are known; however, such methods are not applicable in the case of high dimensionality of the object "curse of dimensionality" and have poor convergence. In this paper, a new approach to solving the problem of program control synthesis based on the use of neural network controller is proposed. The loss function is formulated based on an adapted set of necessary optimality conditions derived from the maximum principle and approximation of the Hamilton-Jacobi-Bellman equation using neural networks. Compared to direct iterative methods, the method presented in this paper can efficiently obtain control from the state-space domain. The demonstration of the proposed method is carried out on a model example of constructing the optimal control of the output to the unstable position of the inverse pendulum, and the results of numerical simulation are given. The research was carried out at the expense of the grant of the Russian Science Foundation No. 23-29-00609, https://rscf.ru/project/23-29-00609.