Towards Globally Optimal Impulsive Spacecraft Rendezvous under a Hovering Constraint a Free Number of Impulses, Free Firing Anomalies, and Three Thrust Regimes

This article studies the minimum-fuel impulsive rendezvous problem about an elliptic reference orbit, in which an active spacecraft must be brought from a given relative state onto a bounded relative trajectory confined indefinitely to a rectangular hovering zone around a passive target. The word "towards" is meant literally and the qualification should be carried throughout: global optimality over all impulse plans is established in full only for the first of the three thrust regimes treated here. For the second, the continuum optimum is bracketed between computed bounds rather than attained; for the third, optimality is proved on the discretisation and not on the continuum. Each claim is stated precisely in the text, and every results table carries the certificate its own regime admits. The starting point is the exact guidance problem: linearised Tschauner-Hempel relative dynamics written in the true anomaly, the anomaly-domain state of Deaconu, a fuel cost that sums the absolute velocity increments axis by axis (an L1 norm, reflecting a fixed thruster configuration), a periodicity condition, a per-impulse saturation bound, and a hovering constraint that must hold at every subsequent anomaly. With the number of impulses fixed and their firing anomalies prescribed, this problem is convex and is solved globally by a single mathematical program. The contribution is to remove both restrictions at once. Neither the number of impulses nor their anomalies is prescribed: the control is sought in the space of vector-valued measures on the manoeuvre window, whose atoms are the impulses. The resulting problem is infinite-dimensional but convex, and its dual is a semi-infinite program in six variables whose constraint bounds Lawden's primer vector. Because the fuel cost is an L1 norm, that bound is componentwise in the thrust axes rather than Euclidean, so each axis carries its own contact condition; in the scenarios considered this is why no radial impulse ever appears. A Caratheodory-type argument bounds the number of impulses in an optimal solution by the dimension of the state, so the impulse count is an output of the theory rather than a modelling choice. An algorithm is proposed combining an exchange method on the dual with an exact Caratheodory reduction of the primal and a verified cardinality reduction. It terminates with matching primal and dual values, which by weak duality proves global optimality over all measures on the window, hence over every impulse count and every choice of firing anomalies simultaneously. The complete method is implemented in portable C around two open-source optimisation libraries, one conic and one linear and mixed-integer. The work is then organised around what the actuator can do, in three regimes treated in turn. With no bound on the thrust the optimum is a small set of impulses and the optimality guarantee is exact. With an upper bound on each thrust component the impulses cannot survive: a bound on the size of an individual impulse is shown not to be closed in the topology that makes the problem well posed, so imposing it on measures changes nothing, and the correct model is a bounded thrust density whose optimal control is bang-off-bang, with finite burn arcs sitting exactly where the primer vector exceeds one. With a lower bound as well - a minimum impulse bit - the admissible set is no longer convex. The lossless-convexification relaxation of that case is shown to be very nearly tight but to describe an engine that can never be switched off, its optimum being a floor independent of the scenario; the physically meaningful reading, in which an axis is either off or firing within its band, is instead stated as a mixed-integer program over semi-continuous variables and solvedexactly. The hovering constraint itself is imposed exactly rather than on a grid. Each face of the zone is, after clearing a strictly positive factor, a trigonometric polynomial in the anomaly, and the Fejer-Riesz theorem represents its non-negativity by a small positive-semidefinite Gram matrix. The representation is an equivalence, not a relaxation, so the semi-infinite constraint is replaced by six semidefinite blocks of size at most six with no loss of optimality, and the scenario-dependent question of how finely to sample disappears. Each regime is demonstrated on four PRISMA-like scenarios, over two orbit families chosen so that perigee altitude stays physically meaningful as the eccentricity varies: a near-circular family at fixed semi-major axis and a transfer-orbit family at fixed perigee. Cost is reported per phase, and the same certificates give, before any impulse is considered, the largest margin by which the zone could be shrunk while still admitting a periodic relative trajectory - so an infeasible scenario is identified as infeasible by geometry rather than mistaken for a solver failure. Every run is certified, by the certificate its own regime admits: a dual measure equal to one where the impulses are unbounded, a matching primal-dual pair where the thrust rate is capped, and a mixed-integer gap of zero, reached in every scenario reported,where the dead band makes the problem nonconvex. The computational conclusion is qualified rather than negative: solves that previously took tens of seconds now take under three, most of them well under one, which narrows but does not close the gap to onboard use, since flight processors are one to two orders of magnitude slower and the iteration count still has no a priori bound.