Optimality Conditions Applied to Free-Time Multiburn Optimal Orbital Transfers

Abstract

While the Pontryagin Maximum Principle can be used to calculate candidate extremals for optimal orbital transfer problems, these candidates cannot be guaranteed to be at least locally optimal unless sufficient optimality conditions are satisfied. In this paper, through constructing a parameterized family of extremals around a reference extremal, some second-order necessary and sufficient conditions for the strong-local optimality of the free-time multi-burn fuel-optimal transfer are established under certain regularity assumptions. Moreover, the numerical procedure for computing these optimality conditions is presented. Finally, two medium-thrust fuel-optimal trajectories with different number of burn arcs for a typical orbital transfer problem are computed and the local optimality of the two computed trajectories are tested thanks to the second-order optimality conditions established in this paper.

A quadratic homotopy method for fuel-optimal low-thrust trajectory design

“…Recently, Chen et al. 22,23 investigated the sufficient and the second-order necessary conditions for the strong-local optimality of the multi-burn fuel-optimal transfer. In their method, the initial energy-optimal problem is solved iteratively with decreasing thrust magnitude, which is actually a homotopy method whose homotopic parameter is embedded into the equations of motion.…”

“…In 2015, Zhang et al 21 employed the linear homotopy method for solving low-thrust minimum-fuel optimization problem in the circular restricted three-body system, which transfers from a geostationary orbit to a halo orbit. Recently, Chen et al 22,23 investigated the sufficient and the second-order necessary conditions for the strong-local optimality of the multi-burn fuel-optimal transfer. In their method, the initial energy-optimal problem is solved iteratively with decreasing thrust magnitude, which is actually a homotopy method whose homotopic parameter is embedded into the equations of motion.…”

A new probability-one homotopy method for solving minimum-time low-thrust orbital transfer problems

“…Homotopy methods, the principle of which is that a given problem is embedded into a family of problems parameterized by a homotopic parameter, and the optimal solution to the original problem is obtained by tracing the optimal solutions of the embedded problems (Watson 2002), have been widely applied to circumvent the above disadvantages of solving low-thrust trajectory optimization problems by single shooting indirect methods. References (Bertrand and Epenoy 2002;Haberkorn et al 2004;Gergaud and Haberkorn 2006;Guo et al 2012;Jiang et al 2012;Zhang et al 2015;Chen 2016;Taheri et al 2016;Chi et al 2017;? ;Pan et al 2018) have successfully utilized homotopy methods to solve minimum-fuel low-thrust orbital transfer problems, the optimal thrusts of which are discontinuous bang-bang controls.…”

“…In these methods, the homotopic parameter is embedded into the performance index to provide continuous transition of optimal controls from the initial problem to the original one. It has been widely observed that the original fuel-optimal lowthrust trajectory optimization problems can be easily solved with probability one once the initial solutions are achieved (Guo et al 2012;Jiang et al 2012;Zhang et al 2015;Chen 2016;Chi et al 2017;Pan et al 2018). In contrast, homotopy methods for solving minimum-time low-thrust orbital transfer problems, whose optimal thrusts keep constant during the whole optimal trajectory, are still not satisfactorily developed.…”