The structure of the Cauchy operator to a linear continuous-discrete functional differential system with aftereffect and some properties of its components

In this paper, a class of linear functional differential systems with aftereffect, continuous and discrete times, and impulses (impulse hybrid systems) is considered. The focus of attention is on the structure of the Cauchy operator to the hybrid system under consideration and the representation of their components. Those allow one to give the representation of all trajectories of the hybrid system and to formulate conditions of the solvability for control problems in various classes of controls, to obtain estimates of the attainability sets under constrained control, and to study general linear boundary value problems for the solvability. A detailed description of all components to the Cauchy operator is given and their properties are studied. For the components with continuous time, some conditions of the continuity with respect to the second argument are obtained which is related to deciding on a class of controls. The main results are based on constructions of the Cauchy matrices to systems with continuous time and difference systems.