On the regularization and matrix Lyapunov functions for fuzzy differential systems with uncertain parameters

Anatoliy Martynyuk; Gani Stamov; Ivanka Stamova; Yulya Martynyuk–Chernienko; Anatoliy Martynyuk; Gani Stamov; Ivanka Stamova; Yulya Martynyuk–Chernienko

doi:10.3934/era.2023310

Electronic Research Archive

2023, Volume 31, Issue 10: 6089-6119. doi: 10.3934/era.2023310

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On the regularization and matrix Lyapunov functions for fuzzy differential systems with uncertain parameters

1.
S. P. Timoshenko Institute of Mechanics, NAS of Ukraine, 3 Nesterov str., Kiev-57, Ukraine
2.
Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio, TX 78249, USA

Received: 31 May 2023 Revised: 30 July 2023 Accepted: 01 September 2023 Published: 12 September 2023

In this paper, for a regularized fuzzy system, a generalization of the direct Lyapunov method is adapted on the base of matrix-valued Lyapunov-like functions. First, the new concept of a regularization scheme for fuzzy systems is discussed and the matrix-valued Lyapunov function technique is introduced. Then, sufficient conditions are established for the boundedness and stability of the equilibrium set of solutions of the regularized fuzzy system of differential equations. Scalar and vector Lyapunov-type functions are used based on an auxiliary matrix-valued function. Finally, a discussion is offered for the future directions of the proposed approach. Since the strategies for the analysis of the stability of fuzzy models are very important in numerous aspects, we expect that our results will inspire researchers to develop the introduced concept.
- fuzzy systems,
- regularization scheme,
- boundedness,
- stability,
- matrix Lyapunov functions
Citation: Anatoliy Martynyuk, Gani Stamov, Ivanka Stamova, Yulya Martynyuk–Chernienko. On the regularization and matrix Lyapunov functions for fuzzy differential systems with uncertain parameters[J]. Electronic Research Archive, 2023, 31(10): 6089-6119. doi: 10.3934/era.2023310

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Abstract

In this paper, for a regularized fuzzy system, a generalization of the direct Lyapunov method is adapted on the base of matrix-valued Lyapunov-like functions. First, the new concept of a regularization scheme for fuzzy systems is discussed and the matrix-valued Lyapunov function technique is introduced. Then, sufficient conditions are established for the boundedness and stability of the equilibrium set of solutions of the regularized fuzzy system of differential equations. Scalar and vector Lyapunov-type functions are used based on an auxiliary matrix-valued function. Finally, a discussion is offered for the future directions of the proposed approach. Since the strategies for the analysis of the stability of fuzzy models are very important in numerous aspects, we expect that our results will inspire researchers to develop the introduced concept.

References

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