On the theory of stability of discontinuous dynamical systems

Authors

  • A.A. Martynyuk S.P. Timoshenko Institute of Mechanics

DOI:

https://doi.org/10.15407/dopovidi2023.01.003

Keywords:

discontinuous systems, matrix-valued Lyapunov functions, stability, La-Salle invariance principle.

Abstract

The theory of differential equations with a discontinuous right-hand side finds application in problems of motion control, in the study of systems with variable structure, in the analysis of automatic control systems with sliding regime and others. The purpose of this article is to present some results of the study of stability obtained for the specified class of systems based on the method of Lyapunov’s matrix-valued functions. These results are formulated in terms of the sign-definiteness of special matrices, which are used to estimate the change in the Lyapunov function and its generalized derivative.

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References

Filippov, A. F. (1985). Differential equations with discontinuous right-hand. Moscow: Nauka (in Russian).

Martynyuk, A. A. (2007). Stability of motion: The role of multicomponent Lyapunov’s functions. Cambridge: Cambridge Scientific Publishers.

Clarke, F. N. (1983). Optimization and nonsmooth analysis. New York: Wiley.

Deriviére, S. & Aziz-Alaoui, M. A. (2006). An invariance principle for discontinuous righthand sides dynamical systems. Res. Reports, Dept. Math. Univ. Sherbrooke, No. 31, pp. 1-13.

Stipanovic, D. M. & Siljak, D. D. (2001, June). Connective stability of discontinuous interconnected systems via parameter-dependent Lyapunov functions. Proceedings of the 2001 American Control Conference (pp. 4189-4196). Arlington, VA. https://doi.org/10.1109/ACC.2001.945633

Martynyuk, A. A. (2004). On stability of motion of discontinuous dynamical systems. Dokl. Acad. Nauk, 397, No. 3, pp. 308-312 (in Russian).

Martynyuk, A. A. & Martynyuk-Chernienko, Ya. A. (2012). Uncertain dynamical systems. Stability and motion control. Boca Raton: CRC Press.

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Published

09.03.2023

How to Cite

Martynyuk, A. . (2023). On the theory of stability of discontinuous dynamical systems. Reports of the National Academy of Sciences of Ukraine, (1), 3–9. https://doi.org/10.15407/dopovidi2023.01.003

Section

Mathematics

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