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

P. M. Simonov

Vestnik rossijskih universitetov. Matematika, Tome 23 (2018) no. 124, pp. 726-737

Résumé

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.

Détail

Keywords: the theorem of Bohl-Perron about the asymptotic stability, hybrid linear system of functional differential equations, method of the model equations, converse theorem.

Bibliographie

[1] V. M. Marchenko, J. J. Luazo, “On the stability of hybrid difference-differential systems”, Differential Equations, 45:5 (2009), 728–740 (In Russian)

[2] P. M. Simonov, “Theorem of Bohl-Perron of hybrid linear functional differential systems with aftereffect”, Bulletin of Perm University. Mathematics. Mechanics. Computer Science, 2016, no. 2(33), 56–60 (In Russian)

[3] P. M. Simonov, “On the question of the theorem of Bohl-Perron of hybrid linear functional differential systems with aftereffect (HLFDSA)”, Middle Volga Mathematical Society Journal, 18:1 (2016), 75–81 (In Russian)

[4] P. M. Simonov, “The Bohl–Perron theorem for hybrid linear systems with aftereffect”, Proceedings of International Symposium “Differential Equations–2016”, Perm, May 17-18, 2016, Itogi Nauki i Tekhniki. Ser. Sovrem. Mat. Pril. Temat. Obz., 132, VINITI, Moscow, 2017, 122–126 (In Russian) | MR | Zbl

[5] P. M. Simonov, “The Bohl–Perron theorem for hybrid linear systems with aftereffect”, Journal of Mathematical Sciences, 230:5 (2018), 775–781

[6] P. M. Simonov, “Theorem of Bohl-Perron on asymptotic stability of hybrid linear functional differential systems with aftereffect (HLFDSA)”, Bulletin of the Russian Academy of Natural Sciences, 16:3 (2016), 55–59 (In Russian)

[7] P. M. Simonov, “The Bohl–Perron theorem on the asymptotic stability of hybrid systems”, Functional and Differentional Equations: Theory and Applications, Proceedings of the Conference Dedicated to the 95th Anniversary of Professor N. V. Azbelev (Perm, 2018), PNIPU, Perm, 2018, 230–235 (In Russian)

[8] N. V. Azbelev, L. M. Berezanskiy, P. M. Simonov, A. V. Chistyakov, “Stability of linear systems with aftereffect. IV”, Differential Equations, 29:2 (1993), 196–204 (In Russian)

[9] N. V. Azbelev, L. M. Berezanskiy, P. M. Simonov, A. V. Chistyakov, “Stability of linear systems with aftereffect. III”, Differential Equations, 27:10 (1991), 1659–1668 (In Russian)

[10] N. V. Azbelev, P. M. Simonov, Stability of Equations Solutions with Ordinary Derivatives, Perm State University, Perm, 2001, 230 pp. (In Russian)

[11] E. A Barbashin, Introduction to the Stability Theory, Nauka Publ., Moscow, 1967, 224 pp. (In Russian)

[12] J. L. Massera, J. J. chaffer, Linear Differential Equations and Function Spaces, Mir Publ., Moscow, 1970, 456 pp. (In Russian)

[13] L. V. Kantorovich, G. P. Akilov, Functional Analysis, BKhV-Peterburg, St. Petersburg, 2004, 816 pp. (In Russian)

[14] V. R. Nosov, “Perron's theorem for stationary and periodic systems of differential functional equations”, Differential Equations with Deviating Argument, 11 (1979), 44–51 (In Russian)

[15] V. B. Kolmanovskiy, V. R. Nosov, Stability and Periodic Regimes Controlled Systems with Aftereffect, Nauka Publ., Moscow, 1981, 448 pp. (In Russian)

[16] V. G. Kurbatov, “On functional differential equations stability”, Differential Equations, 17:6 (1981), 963–972 (In Russian)

[17] V. G. Kurbatov, Linear Differential Equations, Voronezh University Publ., Voronezh, 1990, 168 pp. (In Russian)

[18] V. F. Pulyaev, Z. B. Tsalyuk, “On the issue of the admissibility of certain spaces pairs for linear operators and the Volterra equations”, Differential Equations, 19:4 (1983), 684–692 (In Russian)

[19] V. F. Pulyaev, “On the admissibility of some pairs of spaces according to linear integral Volterra equations”, Differential Equations, 20:10 (1984), 1800–1805 (In Russian)

[20] V. F. Pulyaev, “On the spectrum of linear continuous operators”, Bulletin of the North Caucasus Scientific Center of the Higher School. Natural Science, 1985, no. 4, 25–28 (In Russian)

[21] V. F. Pulyaev, “On the spectrum of Volterra operators”, Integral Operators and the Equation, Proceedings of Scientific Works, Kuban State University, Krasnodar, 1987, 29–37 (In Russian)

[22] V. F. Pulyaev, “On interrelation of Noetherian linear continuous operators and their restrictions”, Russian Mathematics, 339:8 (1990), 65–73 (In Russian)

[23] V. F. Pulyaev, Development of the Theory of Linear Integral Equations with Periodic and Almost Periodic Kernels, Diss. dots Doc. Phys.-Math. Sciences, St. Petersburg, 2001, 31 pp. (In Russian)

[24] V. F. Pulyaev, Z. B. Tsalyuk, “On the asymptotic behavior of solutions of integral Volterra equations in Banach spaces”, Russian Mathematics, 335:12 (1991), 47–55 (In Russian)

[25] D. G. Sokol, “On the admissibility of certain space pairs for integral operators and equations”, University News North-Caucasian Region. Natural Sciences Series, 2000, no. 1, 135–137 (In Russian)

[26] A. V. Bukhvalov, A. I. Veksler, G. Ya. Lozanovskiy, “Banach lattices – some Banach theory aspects”, Russian Mathematical Surveys, 34:2 (1979), 137–183 (In Russian)

[27] V. G. Kurbatov, Functional differential operators and equations, Kluwer Academic Publ., Dordrect, 1999, 433 pp.