Abstract
For standard form nonlinear equations with generalized derivative, estimates of deviation of a set of exact solutions from the averaged ones are established and the deviation of a set of trajectories of averaged equations from the equilibrium state is specified in terms of pseudo-linear integral inequalities. Sets of affine systems and problems of approximate integrations and stability over finite interval are considered as applications.
For standard form nonlinear equations with generalized derivative, estimates of deviation of a set of exact solutions from the averaged ones are established and the deviation of a set of trajectories of averaged equations from the equilibrium state is specified in terms of pseudo-linear integral inequalities. Sets of affine systems and problems of approximate integrations and stability over finite interval are considered as applications.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Bogoliubov, N.N., Mitropolskii, Yu.A.: Asymptotic Methods in Theory of Nonlinear Oscillations. Izd-vo Akademii nauk Ukr. SSR, Moscow (1963)
Grebenikov, E.A., Mitropolsky, Yu.A., Ryabov, Y.A.: Asymptotic Methods on Resonance Analytical Dynamics. Chapmen and Hall/CRC, Boca Raton (2004)
Krylov, N.M., Bogoliubov, N.N.: Introduction to Nonlinear Mechanics. Izd-vo Akademii nauk Ukr. SSR, Kiev (1937)
Lakshmikantham, V., Leela, S., Martynyuk, A.A.: Practical Stability of Nonlinear Systems. World Scientific, Singapore (1990)
Louartassi, Y., El Mazoudi, El.H., El Alami, N.: A new generalization of lemma Gronwall–Bellman. Appl. Math. Sci. 6(13), 621–628 (2012)
Martynyuk, A.A.: Practical Stability of Motion. Naukova Dumka, Kiev (1983)
Martynyuk, A.A.: Stability Analysis: Nonlinear Mechanics Equations. Gordon and Breach Publishers, Philadelphia (1995)
Martynyuk, A.A.: Novel bounds for solutions of nonlinear differential equations. Appl. Math. 6, 182–194 (2015)
Martynyuk, A.A.: Analysis of a set of trajectories of generalized standard systems: averaging technique. Nonlinear Dyn. Syst. Theory 17(1), 29–41 (2017)
Martynyuk, A.A.: Deviation of the set of trajectories from the state of equilibrium. Dokl. NAS Ukr. 10, 10–16 (2017)
Müller, W.: Stability Analysis: Nonlinear Mechanics Equations by Martynyuk A.A. Stability and Control: Theory, Methods and Applications, vol. 2. Gordon and Breach Science Publishers, Philadelphia (1995). Zbl. Math. 0840.34003
Plotnikov, A.V., Skripnik, N.V.: Differential Equations with “Clear” and Fuzzy Multi-Valued Right-Hand Side. Asymptotic Methods. Astroprint, Odessa (2009)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Martynyuk, A.A. (2019). Finite-Time Stability of Standard Systems Sets. In: Qualitative Analysis of Set-Valued Differential Equations. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-07644-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-030-07644-3_8
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-07643-6
Online ISBN: 978-3-030-07644-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)