This paper presents a general purpose neighboring optimal guidance algorithm that is capable of driving a dynamical system along a specified nominal, optimal path. This goal is achieved by minimizing the second differential of the objective function along the perturbed trajectory. This minimization principle leads to deriving all the corrective maneuvers, in the context of a closed-loop guidance scheme. Several time-varying gain matrices, referring to the nominal trajectory, are defined, computed offline, and stored in the onboard computer. Original analytical developments, based on optimal control theory, in conjunction with the use of a normalized time scale, constitute the theoretical foundation for three relevant features: (i) a new, efficient law for the real-time update of the time of flight (the so called time-to-go), (ii) a new termination criterion, and (iii) a new analytical formulation of the sweep method. This new guidance, termed variable-time-domain neighboring optimal guidance, is rather general, avoids the usual numerical difficulties related to the occurrence of singularities for the gain matrices, and is exempt from the main disadvantages of similar algorithms proposed in the past. For these reasons, the variable-time-domain neighboring optimal guidance has all the ingredients for being successfully applied to problems of practical interest.Communicated by Anil Rao.M. Pontani (B) · G. Cecchetti · P. Teofilatto University "La Sapienza",

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In recent years, several countries have shown an increasing interest toward both manned and automatic lunar missions. The development of a safe and reliable guidance algorithm for lunar landing and soft touchdown represents a very relevant issue for establishing a real connection between the Earth and the Moon surface. This paper applies a new, general-purpose neighboring optimal guidance algorithm, proposed in a companion paper and capable of driving a dynamical system along a specified nominal, optimal path, to lunar descent and soft landing. This new closedloop guidance, termed variable-time-domain neighboring optimal guidance, avoids the usual numerical difficulties related to the occurrence of singularities for the gain matrices, and is exempt from the main drawbacks of similar algorithms proposed in the past. For lunar descent, the nominal trajectory is represented by the minimumtime path departing from the periselenium of a given elliptic orbit and arriving at the Moon with no residual velocity. Perturbations arising from the imperfect knowledge of the propulsive parameters and from errors in the initial conditions are considered. At specified, equally spaced times the state displacements from the nominal flight conditions are evaluated, and the guidance algorithm yields the necessary control corrections. Extensive robustness and Monte Carlo tests are performed, 123 J Optim Theory Appl and definitely prove the effectiveness, robustness, and accuracy of the new guidance scheme at hand, also in comparison with the well-established linear tangent steering law.

Multistage launch vehicles are employed to place spacecraft and satellites in their operational orbits. Trajectory optimization of their ascending path is aimed at defining the maximum payload mass at orbit injection, for specified structural, propulsive, and aerodynamic data. This work describes and applies a method for optimizing the ascending path of the upper stage of a specified launch vehicle through satisfaction of the necessary conditions for optimality. The method at hand utilizes a recently introduced heuristic technique, that is, the particle swarm algorithm, to find the optimal ascent trajectory. This methodology is very intuitive and relatively easy to program. The second-order conditions, that is, the Clebsch-Legendre inequality and the conjugate point condition, are proven to hold, and their fulfillment enforces optimality of the solution. Availability of an optimal solution to the second order is an essential premise for the possible development of an efficient neighboring optimal guidance.