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.

Summary. The numerical Solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known integrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.

Second order sufficient conditions (SSC) for control problems with control{state constraints and free final time are presented. Instead of deriving such SSC de initio, the control problem with free final time is tranformed into an augmented control problem with fixed final time for which well-known SSC exist. SSC are then expressed as a condition on the positive definiteness of the second variation. A convenient numerical tool for verifying this condition is based on the Riccati approach where one has to find a bounded solution of an associated Riccati equation satisfying specific boundary conditions. The augmented Riccati equations for the augmented control problem are derived and their modifications on the boundary of the control{state constraint are discussed. Two numerical examples, (1) the classical Earth-Mars orbit transfer in minimal time, (2) the Rayleigh problem in electrical engineering, demonstrate that the Riccati equation approach provides a viable numerical test of SSC.

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