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. 1. Introduction Multistage rockets are employed to place spacecraft and satellites in their operational orbits. The optimization of their ascending trajectory leads to determining the maximum payload mass that can be inserted in the desired orbit. This goal is achieved by finding the optimal control time history and the optimal thrust and coast durations. The numerical solution of aerospace trajectory optimization problems has been pursued with different approaches in the past. Indirect methods, such as the gradient-restoration algorithm [1, 2] and the shooting method [3] or direct techniques, such as direct collocation [4, 5], direct transcription [6, 7], and differential inclusion [8, 9], are examples of such techniques. However, only a relatively small number of publications are concerned with trajectory optimization of multistage launch vehicles [1, 2, 10, 11]. A recently published paper [12] describes a simple method for performance evaluation through generation of a near optimal trajectory for a multistage launch vehicle. This research considers the optimal exoatmospheric trajectory of the upper stage of the latter rocket, whose characteristics are specified. The trajectory arc that precedes orbital injection is composed of two phases: (1) coast (Keplerian) arc and (2) thrust phase. More specifically, for the upper stage the existence and duration of a coast arc (with no propulsion) and the optimal thrust direction are being investigated through the first-order necessary conditions for optimality, that is, the Euler-Lagrange equations and the
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