Singular optimal control of ascent stage for a surface-to-air missile

This paper addresses the Singular Optimal Control Problem (SOCP) for a surface-to-air missile with limited control, fully considering aerodynamic effects with a parabolic drag polar. This problem is an extension of the typical Goddard problem. First, the classical Legendre-Clebsch condition is applied to derive optimal conditions for the singular angle of attack, revealing that the missile turns by gravity along the singular arc. Second, the higher-order differentiation of the switching function provides the necessary conditions to determine the optimal thrust, expressed as linear functions of the costate variables. The vanishing coefficient determinant is then employed to decouple the control and costate variables, yielding the singular thrust solely dependent on state variables and identifying the singular surface. Moreover, the analytical singular control can be regarded as path constraints subject to the typical Optimal Control Problem (OCP), enabling the GPOPS-Ⅱ, a direct method framework that does not involve the singular condition, to solve the SOCP. Finally, three cases with different structures are presented to evaluate the performance of the proposed method. The results show that it takes a few steps to obtain the numerical optimal solution, which is consistent with the analytical solution derived from the calculus of variations, highlighting its great computational accuracy and effectiveness.