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

Abstract

A new method to find the wave functions and binding energy of the three nucleon system is discussed. Results of the calculations with local, central potentials are given. The wave functions are found to be sensitive to the shape of the nucleon-nucleon potential. The corresponding binding energies differ by less than 0.6 MeV.

Optimality Conditions Applied to Free-Time Multiburn Optimal Orbital Transfers

“…This equation generalizes the second-order condition for fixed-time problems in [24] to the problems with free final time. Then, as a result of Corollary 1, we eventually obtain the following theorem.…”

“…Let (x(t, p 0 ), p(t, p 0 )) = γ(t, p 0 ) for (t, p 0 ) ∈ [0, t f ] × P and assume that the final time is fixed; a fold singularity occurs at a time t c ∈ (0,t f ] if det [∂x(t c ,p 0 )/∂p 0 ] = 0 [22]. Hence, conjugate points for fixed-time orbital transfer problems are tested by detecting the zero of det [∂x(·,p 0 )/∂p 0 ] on (0,t f ] in [23,24,26]. However, according to Eq.…”

Neighboring optimal control for fixed-time multi-burn orbital transfers

“…If the matrix ∂x ∂q (t,q) is singular at a time t ∈ (t i , t i+1 ), the projection Π of the family F at t is a fold singularity [21][22][23][24].…”

“…If the subset N is small enough, this condition guarantees that projection Π of the family F on each subinterval (t i , t i+1 ) for i = 0, 1, · · · , k with t k+1 = t f is a diffeomorphism, see Refs. [21][22][23]. However, this condition is not sufficient to guarantee the projection Π of the family F on the whole semi-open interval [t 0 , t f ) is a diffeomorphism because there exists another type of fold singularity near each switching time t i , as is illustrated by Figure 3 that the trajectories x(t, q) around the switching time t i (q) may intersect with each other [20,21].…”

“…Then, as a result of a geometric study on the projection of the parameterized family from tangent bundle onto state space, it is presented in this paper that the conditions sufficient for the existence of neighboring extremals around a bangbang extremal consist of not only the JC between switching times but also a TC [20][21][22] at each switching time. According to recent advances in geometric optimal control [21][22][23][24], the JC and the TC, once satisfied, are also sufficient to guarantee the nominal extremal to be locally optimal provided some regularity assumptions are satisfied. Given these two existence conditions, the neighboring optimal feedbacks on thrust direction and switching times are established in this paper through deriving the first-order Taylor expansion of the parameterized neighboring extremals.…”