Use of finite-dimensional approximations in a problem of stabilization of periodic systems with aftereffect

We consider a stabilization problem for linear periodic system of differential equations with aftereffect. Approximating systems are described by differential equations with finite-dimensional Volterra operators. We construct admissible controls by feedback principle in a class of piecewise continuous functions. We obtain a relation between approximating problem of stabilization and a problem of optimal stabilization of autonomous linear system of difference equations.