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Summary. A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the weltknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically.Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.

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Summary.A modification of the well-known continuation (or homotopy) method for actual computation is worked out. Compared with the classical method, the modification seems to be a more reliable device for supplying useful initial data for shooting techniques. It is shown that computing time may be significantly reduced in the numerical solution of sensitive realistic two-point boundary value problems.

SUMMARYOn the basis of Part 1 of this paper, a numerical method is developed for the real-time computation of neighbouring optimal feedback controls for constrained optimal control problems. We use the idea of multiple shooting to develop a numerical method which has the following properties:1. The method is applicable to optimal control problems with constraints (differential equations, boundary conditions, inequality constraints, problems with discontinuities, etc.). 2. The control variables and the switching points are computed for the remaining time interval of the process. 3. All constraints are checked. 4. The method is appropriate for real-time computations on onboard computers of space vehicles. 5 . The scheme is robust in that controllable deviations from a precalculated flight path are much largerThe re-entry of a space vehicle is investigated as an example. One problem contains a control variable inequality constraint with a large variety of different switching structures, including problems with a corner. A second problem contains a state variable inequality constraint with one or two boundary points or one boundary arc. The different switching structures depend on the tightness of the constraints.KEY WORDS Numerical methods Multiple shooting Repeated correction method Real-time computation than deviations typical for perturbations occuring in space vehicles. Guidance of space vehicles