Decomposition of a singularly perturbed functional-differential system based on a nondegenerate transformation

For a linear, stationary, singularly perturbed functional-differential control system with a small coefficient of the highest derivative and a finite delay in slow state variables, we justify the decomposition by a nondegenerate transformation of variables, which is a generalization of the well-known Chang transformation. This transformation splits the original two-rate system into two independent subsystems of lower dimension separately for the fast and slow variables. We prove that the splitting transformation can be constructed with any approximation accuracy in the form of an asymptotic expansion in powers of a small parameter and propose an iterative scheme for calculating the asymptotic series. Based on the decomposition constructed, we establish that for sufficiently small values of the parameter, the spectrum of the system is split into two sets, separately for “small” and “large” eigenvalues. Also, we give examples of constructing transform approximations.