Review of homotopy methods for aerospace trajectory optimization

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.