Asymptotic approximations validity boundaries for decoupling transformation of three-time-scale linear time-invariant singularly perturbed systems with delay

For time-invariant singularly perturbed control systems with state delay the method of separation of movements is evolved on the basis of Chang-type non-degenerate transformation. Asymptotic approximations for completely separated subsystems of the considered singularly perturbed system with three-time scales are introduced, boundaries of values of small singularity parameters are constructed and proved, which guarantee the validity of asymptotic representations and estimates of solutions underlying matrix operator equations, asymptotic approximations for the decoupling transformation and matrix operators of the split system. An illustrative example is given.