Asymptotics of the splitting transformation for a linear stationary singularly perturbed system with delay

The splitting transformation is a generalization of the well-known Chang transformation for linear, stationary, singularly perturbed system with many delays in slow-state variables; it reduces the original two-speed system to two independent subsystems of smaller dimensions with different rates of change of variables. The splitting transformation leads us to Riccati and Sylvester equations for functional matrices, which can be found in the form of asymptotic series in powers of the small parameter. In this work, we prove that asymptotic approximations of any order of accuracy based on these series can be represented as finite sums in powers of $\lambda$. We compare exact solutions with approximations obtained by the method proposed.