Conditions for pointwise controllability and pointwise observability of linear time-invariant singularly perturbed systems with delay

For a linear time-invariant singularly perturbed system with finite lumped delay in state variables and with an observable linear output without delay, multipoint finite-dimensional boundary value problems are considered - the problems of pointwise controllability and pointwise observability. The duality of the considered singularly perturbed control and observation systems is established. By the method of defining equations, necessary, sufficient conditions for pointwise controllability and pointwise observability, which are valid for all sufficiently small values of the singularity parameter, are obtained independent of the parameter. The conditions are expressed in terms of the matrix parameters of the original system, have a rank type, and are formulated in terms of solutions of recurrent algebraic matrix defining equations, which are constructed according to the original systems by simple rules.