Necessary conditions for the optimality of singular arcs of spacecraft trajectories subject to multiple gravitational bodies

$$\mathrm{L}^1$$ L 1 -optimality conditions for the circular restricted three-body problem

“…[9], Corollary 1 shows an insight into the control structure of the L 1 -minimization trajectory, i.e., there exists a chattering arc when concatenating a singular arc with a nonsingular arc. The chattering arcs may not be found by direct numerical methods when concatenating singular arcs with nonsingular arcs [10].…”

On properties of optimal controls for an inverted spherical pendulum

“…Singular modes are characterized by the fact that over some open interval the Hamiltonian reaches a maximum at more than one point, that is, the optimal control is not determined directly from the maximality condition of the Pontryagin maximum principle. Singular solutions appear in many applications: optimal spacecraft flights (intermediate thrust arcs) or problems of spacecraft reorientation [15][16][17][18][19][20], robotics (controlling manipulators [21,22], the Dubins car problem [23]), mathematical models in economics [24,25], biomedical problems [26,[28][29][30]. For more details on singular solutions see, for example, [22,31,32].…”

Optimal spiral-like solutions near a singular extremal in a two-input control problem

“…Singular modes (or singular regimes) are characterized by the fact that over some finite time interval the Hamiltonian reaches a maximum at more than one point, that is, the optimal control is not determined directly from the maximality condition of the Pontryagin maximum principle. Singular solutions appear in many applications: optimal spacecraft flights (intermediate thrust arcs) or problems of spacecraft reorientation [20,36,40,43,57,58], robotics (controlling manipulators [51,30], the Dubins car problem [1], a mobile robot [39]), mathematical models in economics [53,48], biomedical problems [25,26,49]. For more details on singular solutions see, for example, [10,24,51].…”