Homotopy methods have received increasing attention for their effectiveness in solving nonlinear optimal control problems, particularly in the aerospace domain. This review provides a comprehensive survey of the development and application of homotopy methods for trajectory optimization, with a particular emphasis on recent advances. The methodological evolution and theoretical foundations of homotopy are first outlined. Subsequently, homotopy methods within the optimal control framework are presented according to the indirect and direct approaches. Based on the scope of influence of the homotopy mapping, these methods are further categorized into locally structured and globally structured approaches, with their respective characteristics emphasized. Finally, the key open challenges are synthesized to elucidate the inherent limitations of current homotopy approaches and to highlight research directions that may further unlock their potential in aerospace trajectory design and optimization.

In view of the heat flow, overload, and dynamic pressure constraint problems during the reentry process of the lift reentry vehicle, a calculation method of the reentry reachability region under multiple constraints is designed. Using the virtual target method, an optimal control model for the reentry reachability region is developed without the state inequality restrictions. A predictive correction method is designed based on the optimal control without process constraints, which converts state inequality constraints into control inequality constraints. The numerical simulation of the X-33 is completed. The results of the numerical simulation demonstrate that, in contrast to the conventional "soft constraint" method, which depends on the quasi-equilibrium glide conditions, the suggested method is capable of achieving “hard constraint,” meaning that all process constraints can still be strictly satisfied even in the event that the aircraft makes large maneuvers quickly.