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| As is well known in finite dimensions, the peculiarity of $L^1$-minimization is the sparsity of solutions. In the infinite dimensional setting of optimal control, this typically results in subarcs with zero control. An important application is the case of systems with varying mass; when this variation, corresponding to propellant consumption, is proportional to the control norm, minimization of the consumption is equivalent to minimizing the $L^1$-norm of the control. Such problems were investigated in the 60's looking for minimum fuel orbit transfers in space mechanics. Among others, Robbins chiefly addressed the existence and properties of second order singular arcs in the two-body controlled problem. In particular, fuel minimization exhibits the Fuller phenomenon. This was extensively studied by Marchal, then more recently by Zelikin and Borisov.
Building upon the work of Simo et al, some modern approaches use the dynamical properties of the three body problem to devise low cost trajectories. A purely optimal control approach combining single shooting and homotopy method is used here to compute minimum fuel trajectories of the 2BP and 3BP. On the basis of ideas of Schattler, an analysis of the local optimality of these trajectories is proposed. |
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