An Efficient Method for Singular Optimal Control of Goddard Problem Based on Radau Pseudospectral Method

This paper proposes an efficient method for the Goddard problem based on the Radau pseudospectral method. Firstly, by supplementing the necessary conditions, the mathematical expressions for the singular thrust and singular surface are derived, effectively overcoming the theoretical limitations of the calculus of variations in singular optimal control problem analysis. Secondly, after preliminarily determining the structure of the optimal control using the pseudospectral method, the singular thrust is transformed into the path constraint for a secondary precise solution, successfully addressing the numerical instability issues encountered by the traditional pseudospectral method when solving the Goddard problem. Finally, numerical verification is performed. The result shows that the proposed method significantly reduces the computational error of the constant zero Hamiltonian function, improves computational efficiency, and further decreases the performance index, fully validating the effectiveness of the method.