By Multiple Shooting

The multiple shooting method has proved to be a very reliable and eecient method for the solution of multipoint boundary value problems such as those obtained from the necessary conditions for the solutions of constrained optimal control problems. If this technique is also to be used for guidance purposes, special features have to be implemented to reach a computing speed which enables real-time computations. For this purpose, a key problem is the fast approximation of the so-called fundamental or transition matrices whose computation usually requires a tremendous amount of computing time for the required numerical integration. A fourth order Hermite interpolation procedure can serve to approximate these matrices without any integration using a representation of the transition matrices by Volterra-product integrals. This representation allows an easy computation of certain derivatives of the transition matrices. Moreover, special emphasis is laid on the relation between this \optimal" method and two diierent approaches based on a linearization technique yielding only nearly optimal feedback controls. This drawback is more than compensated by the fact that these methods either do not need any numerical integration or a single integration of the equations of motion, only, and neither a tedious integration of adjoint diierential equations nor an iteration process is required. A regularity condition inherent for the multiple shooting algorithm is shown to be suucient for the controllability independent on which of these methods is used. The domain of convergence of Newton's method in multiple shooting is of the same size as the controllability tubes obtained from these neighboring optimum feedback schemes. Numerical examples are discussed for an optimal deceleration maneuver of an Apollo-type vehicle and for a maximum crossrange maneuver of a Space-Shuttle vehicle, both in the Earth atmosphere.