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.

This paper proposes an optimal midcourse guidance method for dual pulse air-to-air missiles, which is based on the framework of the linear Gauss pseudospectral model predictive control method. Firstly, a multistage optimal control problem with unspecified terminal time is formulated. Secondly, the control and terminal time update formulas are derived analytically. In contrast to previous work, the derivation process fully considers the Hamiltonian function corresponding to the unspecified terminal time, which is coupled with control, state, and costate. On the assumption of small perturbation, a special algebraic equation is provided to represent the equivalent optimal condition for the terminal time. Also, using Gauss pseudospectral collocation, error propagation dynamical equations involving the first-order correction term of the terminal time are transformed into a set of algebraic equations. Furthermore, analytical modification formulas can be derived by associating those equations and optimal conditions to eliminate terminal error and approach nonlinear optimal control. Even with their mathematical complexity, these formulas produce more accurate control and terminal time corrections and remove reliance on task-related parameters. Finally, several numerical simulations, comparisons with typical methods, and Monte Carlo simulations have been done to verify its optimality, high convergence rate, great stability and robustness.