On the Stability of Tubes of Discontinuous Solutions
of Bilinear Systems with Delay
The paper considers the stability property of tubes of discontinuous solutions of a bilinear system with a generalized action on the right-hand side and delay. A feature of the system under consideration is that a generalized (impulsive) effect is possible non-unique reaction of the system. As a result, the unique generalized action gives rise to a certain set of discontinuous solutions, which in the work will be called the tube of discontinuous solutions.The concept of stability of discontinuous solutions tubes is formalized. Two versions of sufficient conditions for asymptotic stability are obtained. In the first case, the stability of the system is ensured by the stability property of a homogeneous system without delay; in the second case, the stability property is ensured by the stability property of a homogeneous system with delay. These results generalized the similar results for systems without delay.
Alexander Sesekin, Dr. Sci. (Phys.–Math.), Prof., Ural Federal University, 19, Mira St., Ekaterinburg, 620002, Russian Federation, tel.: (343)375-41-40; Leading Researcher, N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Division of Russian Academy of Sciences, 16, Kovalevskay st., Ekaterinburg, 620990, Russian Federation, e-mail: email@example.com
Natalia Zhelonkina, Senior Lecturer, Ural Federal University, 19, Mira St., Ekaterinburg, 620002, Russian Federation, tel.: (343)375-41-40, e-mail: firstname.lastname@example.org
Sesekin A.N., Zhelonkina N.I. On the Stability of Tubes of Discontinuous Solutions of Bilinear Systems with Delay. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 31, pp. 96-110. https://doi.org/10.26516/1997-7670.2020.31.96
1. Bellman R. Stability Theory of Differential Equations. Dover Books on Mathematics, 2008.
2. Dykhta V.A., Samsonyuk O.N. Optimum impulse control with applications. Moscow, Fizmatlit Publ., 2000. (in Russian)
3. Krasovsky N.N. Theory of motion control. Linear system. Moscow, Nauka Publ.,1968. (in Russian)
4. Miller B.M., Rubinovich E.Ya. Discontinuous solutions in the optimal control problems and their representation by singular space-time transformations. Automation and Remote Control, 2013, vol. 74, pp.1969-2006. https://doi.org/10.1134/S0005117913120047
5. Sesekin A.N. The properties of the attainability set of a dynamical system with impulse control. Automation and Remote Control, 1994, vol. 55, no. 2, pp. 190-195.
6. Sesekin A.N. On sets of discontinuous solutions of nonlinear differential equations. Russ. Math., 1994, vol.38, no. 6, pp.81–87.
7. Sesekin A.N., Fetisova Yu.V.. Functional Differential Equations in the Space of Functions of Bounded Variation. Proceeding of the Steklov Institute of Mathematics. 2010, vol. 269, suppl. 2. pp. 258–265. https://doi.org/10.1134/S0081543810060210
8. Sesekin A.N., Zhelonkina N.I. On the stability of linear systems with generalized action and delay. IFAC-PapersOnLine, Proceedings of the 18th IFAC World Congress Milano, Italy, 2011, pp.13404-13407. https://doi.org/10.3182/20110828-6-IT-1002.02426
9. Sesekin A.N., Zhelonkina N.I. Tubes of Discontinuous Solutions of Dynamical Systems and Their Stability. AIP. Conference Proceeding, 2017, vol.1895, pp.050011 1-7. https://doi.org/10.1063/1.5007383
10. Zavalishchin S.T., Sesekin A.N. Dynamic Impulse Systems: Theory and Applications. Dordrecht, Kluwer Academic Publishers, 1997.