Geodesic
The theorem of Bohl-Perron on the asimptotic stability of hybrid systems and inverse theorem
Vestnik rossijskih universitetov. Matematika, Tome 23 (2018) no. 124, pp. 726-737

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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.
Keywords: the theorem of Bohl-Perron about the asymptotic stability, hybrid linear system of functional differential equations, method of the model equations, converse theorem. 
@article{VTAMU_2018_23_124_a16,
     author = {P. M. Simonov},
     title = {The theorem of {Bohl-Perron} on the asimptotic stability of hybrid systems and inverse theorem},
     journal = {Vestnik rossijskih universitetov. Matematika},
     pages = {726--737},
     publisher = {mathdoc},
     volume = {23},
     number = {124},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VTAMU_2018_23_124_a16/}
}
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P. M. Simonov. The theorem of Bohl-Perron on the asimptotic stability of hybrid systems and inverse theorem. Vestnik rossijskih universitetov. Matematika, Tome 23 (2018) no. 124, pp. 726-737. http://geodesic.mathdoc.fr/item/VTAMU_2018_23_124_a16/