Abstract
We introduce the notions of Poisson total boundedness of solutions, partial Poisson total boundedness of solutions, and partial Poisson total boundedness of solutions with partly controlled initial conditions. We use the Lyapunov vector function method to obtain sufficient conditions for the Poisson total boundedness of solutions, the partial Poisson total boundedness of solutions, and the partial Poisson total boundedness of solutions with partly controlled initial conditions. As a consequence, we obtain sufficient conditions for the above-mentioned kinds of Poisson total boundedness of solutions based on the Lyapunov function method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. Yoshizawa, “Liapunov’s function and boundedness of solutions,” Funkcial. Ekvac. 2, 95–142 (1959).
V. V. Rumyantsev and A. S. Oziraner, Stability and Stabilization of Motion with Respect to Part of the Variables (Nauka, Moscow, 1987) [in Russian].
K. S. Lapin, “Ultimate boundedness with respect to part of the variables of solutions of systems of differential equations with partly controlled initial conditions,” Differ. Uravn. 49 (10), 1281–1286 (2013) [Differ. Equations 49 (10), 1246–1251 (2013)].
K. S. Lapin, “Uniform boundedness in part of the variables of solutions to systems of differential equations with partially controllable initial conditions,” Mat. Zametki 96 (3), 393–404 (2014) [Math. Notes 96 (3), 369–378 (2014)].
K. S. Lapin, “Partial uniform boundedness of solutions of systems of differential equations with partly controlled initial conditions,” Differ. Uravn. 50 (3), 309–316 (2014) [Differ. Equations 50 (3), 305–311 (2014)].
V. M. Matrosov, Vector Lyapunov Functions Method: Nonlinear Analysis of Dynamical Properties (Fizmatlit, Moscow, 2000) [in Russian].
K. S. Lapin, “Partial total boundedness of solutions to systems of differential equations with partly controlled initial conditions,” Mat. Zametki 99 (2), 239–247 (2016) [Math. Notes 99 (2), 253–260 (2016)].
K. S. Lapin, “Lyapunov vector functions and partial boundedness of solutions with partially controlled initial conditions,” Differ.Uravn. 52 (5), 572–578 (2016) [Differ. Equations 52 (5), 549–566 (2016)].
V. I. Vorotnikov and Yu. G. Martyshenko, “On partial stability theory of nonlinear dynamic systems,” Izv. Ross. Akad. Nauk, Teor. Sist. Upravl. 2010 (5), 23–31 (2010) [J. Comput. Syst. Sci. Int. 49 (5), 702–709 (2010)].
V. I. Vorotnikov and Yu. G. Martyshenko, “On problems of partial stability for delay systems,” Trudy Inst. Mat. Mekh. UrO RAN 19 (1), 49–58 (2013).
V. I. Vorotnikov and Yu. G. Martyshenko, “Stability in part of the variables of ‘partial’ equilibria of systems with aftereffect,” Mat. Zametki 96 (4), 496–503 (2014) [Math. Notes 96 (4), 477–483 (2014)].
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations (Gostekhizdat, Moscow, 1947) [in Russian].
K. Miki, A. Masamichi, and S. Shoichi, “On the partial total stability and partially total boundedness of a systemof ordinary differential equations,” Res. Rept. Akita Tech. Coll. 20, 105–109 (1985).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © K.S. Lapin, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 2, pp. 243–254.
Rights and permissions
About this article
Cite this article
Lapin, K.S. Poisson Total Boundedness of Solutions of Systems of Differential Equations and Lyapunov Vector Functions. Math Notes 104, 253–262 (2018). https://doi.org/10.1134/S000143461807026X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S000143461807026X