Two topics in the theory of optimal trajectory analysis

The work presented in this thesis is concerned with certain problems in the domain of optimal trajectory analysis. The general problem in this field is to determine the ‘best’ way to use a certain travel vehicle to carry out some particular mission. Thus our vehicle may be a rocket which is designed to operate with constant exhaust velocity or at constant power. Our mission may be to effect a transfer from the orbit of the Earth to the orbit of some destination planet within the solar system. And our criterion, for judging which manner of utilizing our travel vehicle is ‘best’ may be the requirement that the fuel consumption or the transit time be minimized. The work which follows consists basically of three parts. In Part I the well known problem in the Calculus of Variations known as the Mayer Problem is outlined, since this is the basic mathematical tool which will be used in Parts II and III. In Part II an investigation is made of the thrust programmes obtained when the thrust characteristic of a rocket is of a more general nature than the constant exhaust velocity thrust characteristic which, for instance, is assumed throughout in [1]. It is shown how well known results for the case of a rocket operating at constant exhaust velocity may be obtained as limits of the more general results obtained in that part. Also, the case of optimization for a rocket operating at constant power is considered as a special case and is solved completely for the one dimensional case. In Part III, the travel vehicle considered is the so-called ‘solar’ sail’. The thrust tor this vehicle is derived from the radiation pressure of the solar radiation reflected from the sail surface. For a vehicle of this type, the question of minimizing fuel consumption does not arise once the vehicle has been assembled in space, so that the payoff criterion of most interest will be minimization of transit time. A more detailed conspectus of the problems considered and the results obtained is presented in the separate introductions to Parts II and III