The theorem of Bohl-Perron on the asimptotic stability of hybrid systems and inverse theorem

We consider an abstract hybrid system of two equations with two unknowns: a vector function $x$ that is absolutely continuous on each finite interval $[0,T],$ $T > 0,$ and a sequence of numerical vectors $y.$ The study uses the $W$-method N.V. Azbelev. As a model, a system containing a functional differential equation with respect to $x$ is used, and a difference equation with respect to $y.$ Solutions spaces are studied. For a hybrid system, the Bohl–Perron theorem on asymptotic stability and the converse theorem are obtained.