A Survey of Certain Methods for the Guidance of Space Vehicles

The practical applicability of optimal solutions of control problems substantially depends upon the availability of feedback control schemes to compensate for unavoidable disturbances. Three numerical methods of treating these schemes are presented and discussed in this paper. They can be applied to a rather general class of optimal control problems (for example, problems with inequality constraints and discontinuities). The evaluation of the necessary conditions of optimal control theory yields a so-called multi-point boundary value problem with jump conditions of the form: _ z(t) = h(z(t)) for t j < t < t j+1 and j = 1;. Here the vector-valued function z comprises the state variables and the adjoint variables originating from the necessary conditions. The right hand side of (1) is piecewisely deened according to the assumed switching structure of the problem. The multiple-shooting method (e.g. Bulirsch 1], Stoer and Bulirsch 6] and Oberle 3]) is especially appropriate for handling this type of problem. The concept of neighboring extremals is a most promising approach to construct so-called neighboring optimum feedback schemes. This approach was recently generalized to this rather general class of optimal control problem yielding a linear multipoint boundary value problem (see 4]). For perturbed boundary conditions, which is the case of most practical interest, this boundary value problem can be written as: _ y(t) = T(t) y(t) for t j < t < t j+1 and j = 1; .