AO - Line and On - Line Computationof Optimal Trajectories in

B Abstract. The present paper is an introductory and survey paper of the treatment of realistically modelled optimal control problems from applications in the aerospace eld. Especially those problems are considered which include diierent types of constraints. In the tutorial part of the paper, recipes are given for the treatment of optimal control problems for which, among other constraints, control and/or state variable inequality constraints are to be taken into account. Optimal control problems having singular subarcs and/or discontinuities are also investigated. The discussion of the necessary conditions aims to the subsequent application of the multiple shooting method, which is known to be a very precise and eecient method for the solution of those multipoint boundary-value problems that arise from these necessary conditions. Homotopy techniques as well as the fusion of direct collocation and multiple shooting techniques are described. Both approaches facilitate the construction of an appropriate initial trajectory to start the multiple shooting iteration. In the survey part of the paper, some recently published new guidance methods are described. These methods are based, on the one hand, on the theory of neighboring extremals and, on the other hand, on the multiple shooting method. They are designed for the real-time computation of optimal trajectories in aerospace applications. Five challenging optimal control problems from the eld of aerospace engineering run throughout the whole paper and illustrate the use of the recipes, i.e., the necessary conditions that must be satissed by an optimal solution. Numerical results are given for these problems to demonstrate the performance of the multiple shooting method for the oo-line computation of optimal trajectories as well as the performance of the guidance schemes for the on-line computation of optimal trajectories.