Minimal realization of averaging approximations by averaging approximations

For the averaging approximating systems Σ^N of the controlled linear retarded oscillator in the state space R²×L², minimal realizations are constructed that can be interpreted as averaging approximations of another delay control system, again in a state space with structure R²×L². A state transformation from Σ^N to its minimized realizations is derived, which allows the application of the latter to approximating optimal control problems posed subject to Σ^N , making it possible to economize on time and storage without loss of accuracy