PHYSICS-INFORMED NEURAL NETWORKS APPLIED TO A SERIES OF CONSTRAINED SPACE GUIDANCE PROBLEMS

The newly developed method named Extreme Theory of Functional Connections, or X-TFC, is exploited in this paper to solve constrained optimal control problems. This framework belongs to the family of Physics-Informed Neural Networks (PINNs), and it exploits the so-called Constrained Expressions (CEs) to approximate the latent (unknown) solutions. These expressions, developed within the Theory of Functional Connections (TFC) framework, are the sum of a freechosen function, and a functional that always analytically satisfies the boundary conditions. According to the X-TFC method, the free function is a single layer neural network trained via Extreme Learning Machine (ELM) algorithm. Optimal control problems are treated via indirect method, based on the Hamiltonian of the problem and the Pontryagin Minimum Principle to obtain the optimal control and the first order necessary conditions. Within this formulation, inequality constraints are considered by introducing new variables and additional terms in the cost function, and in the Hamiltonian. Moreover, saturation functions are used to consider the boundaries of inequality constraints. X-TFC is then employed to solve the boundary value problem that arises from the indirect method. Since the boundary conditions are a priori satisfied, accurate results are obtained with a low computational time.