Abstract
Certain sufficient criteria for the types of partial boundedness of solutions with partially controllable initial conditions are obtained in terms of higher-order derivatives of the Lyapunov functions.
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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 ofMotion with Respect to some of the Variables (Nauka, Moscow, 1987) [in Russian].
V. I. Vorotnikov, “On the stability and the stability with respect to part of the variables of “partial” equilibrium positions of nonlinear dynamical systems,” Dokl. Ross. Akad. Nauk 389 (3), 332–337 (2003) [Dokl. Phys. 48 (3), 151–155 (2003)].
V. I. Vorotnikov and Yu. G. Martyshenko, “On the problem of the partial detectability of nonlinear dynamical systems,” Avtomat. Telemekh., No. 1, 25–38 (2009) [Autom. Remote Control 70 (1), 20–32 (2009)].
V. I. Vorotnikov and Yu. G. Martyshenko, “On the theory of partial stability of nonlinear dynamical systems,” Izv. Ross. Akad. Nauk Teor. Sist. Upr., No. 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 systems with aftereffect,” Trudy Inst. Mat. Mekh. UrORAN 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 (3–4), 477–483 (2014)].
E. V. Shchennikova, “Quasistable properties of the solutions of a multiply connected system of differential equations,” Mat. Zametki 91 (1), 136–142 (2012) [Math. Notes 91 (1), 128–134 (2012)].
A. V. Shchennikov, “Inclusion principle and quasistable properties of “partial” equilibrium of dynamical systems,” Vestnik St. PetersburgUniv. Ser. 10, Prikl. Matem. Inform. Prots. Upravl., No. 4, 119–132 (2011).
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)].
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 Differ. Uravn. 49 (10), 1246–1251 (2013)].
E. V. Shchennikova, “Lyapunov functions and ultimate boundedness with respect to part of the variables,” Matem. Modelirovanie 9 (10), 24 (1997).
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 (3–4), 369–378 (2014)].
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 (1–2), 253–260 (2016)].
R. Z. Abdullin, L. Yu. Anapol’skii, A. A. Voronov, A. S. Zemlyakov, R. I. Kozlov, A. I. Malikov, and V. M. Matrosov, The Method of Vector Lyapunov Functions in Stability Theory (Nauka, Moscow, 1987) [in Russian].
V. M. Matrosov, Method of Vector Lyapunov Functions: Analysis of the Dynamic Properties of Nonlinear Systems (Fizmatlit, Moscow, 2001) [in Russian].
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Original Russian Text © K. S. Lapin, 2017, published in Matematicheskie Zametki, 2017, Vol. 101, No. 6, pp. 883–893.
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Lapin, K.S. Higher-order derivatives of Lyapunov functions and partial boundedness of solutions with partially controllable initial conditions. Math Notes 101, 1000–1008 (2017). https://doi.org/10.1134/S0001434617050273
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DOI: https://doi.org/10.1134/S0001434617050273