### Theory of Fuzzy Differential Equations and Inclusions (Mathematical Analysis and Applications)

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Differential inclusions arise in many situations including differential variational inequalities , projected dynamical systems , dynamic Coulomb friction problems and fuzzy set arithmetic.

## Partial Averaging of Fuzzy Hyperbolic Differential Inclusions :: Science Publishing Group

Existence of solutions for the initial value problem. Uniqueness of solutions usually requires other conditions. Then the initial value problem. Differential inclusions can be used to understand and suitably interpret discontinuous ordinary differential equations, such as arise for Coulomb friction in mechanical systems and ideal switches in power electronics.

An important contribution has been made by A. Filippov, who studied regularizations of discontinuous equations. Further the technique of regularization was used by N. Krasovskii in the theory of differential games. Differential inclusions are also found at the foundation of non-smooth dynamical systems NSDS analysis, [2] which is used in the analog study of switching electrical circuits using idealized component equations for example using idealized, straight vertical lines for the sharply exponential forward and breakdown conduction regions of a diode characteristic [3] and in the study of certain non-smooth mechanical system such as stick-slip oscillations in systems with dry friction or the dynamics of impact phenomena.

From Wikipedia, the free encyclopedia. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. One can similarly define corresponding notion for the system It holds that.

Then practical stability properties of system 22 imply the corresponding h 0 , h -practical stability properties of Next we prove that system 18 is h 0 , h -strongly practically stable. Hence system 18 is h 0 , h -strongly practically stable.

Finally, we show that system 18 is h 0 , h -practically asymptotically stable. Now, let us suppose that 22 is practically asymptotically stable. Practical asymptotic stability of 22 means that. Suppose that the conditions of Theorem 21 are satisfied except that condition A 3 is replaced by. Then Q 0 , Q -practical stability properties of system 22 imply the corresponding h 0 , h -practical stability properties of the system Assume that 22 is Q 0 , Q -practically stable. Suppose that the h 0 , h -practical stability of 18 does not hold; then, arguing as in Theorem 16 , by A 2 , A 6 , we can show that.

### 1st Edition

The authors would like to thank the reviewers for their valuable suggestions and comments. The authors declare that there is no conflict of interests regarding the publication of this paper. All authors completed the paper together. All authors read and approved the final paper. National Center for Biotechnology Information , U. Journal List ScientificWorldJournal v.

Published online Jan Author information Article notes Copyright and License information Disclaimer. Received Nov 8; Accepted Dec Wang and W. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract We present a new comparison principle by introducing a notion of upper quasi-monotone nondecreasing and obtain the practical stability criteria for set valued differential equations in terms of two measures on time scales by using the vector Lyapunov function together with the new comparison principle. Introduction Stability theory in the sense of Lyapunov is now well known.

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Remark 9 — The usual scalar subtraction occurs in the scalar coefficient at the beginning of the above expressions and the Hukuhara set difference occurs inside the brackets. Theorem 16 — Assume that H 1. Then one says that i. Other practical stability notions can be defined similarly. Theorem 21 — Assume that A 1. Theorem 22 — Suppose that the conditions of Theorem 21 are satisfied except that condition A 3 is replaced by A 6. Proof — Assume that 22 is Q 0 , Q -practically stable. Acknowledgments The authors would like to thank the reviewers for their valuable suggestions and comments.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

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Authors' Contribution All authors completed the paper together. References 1. Lasalle JP, Lefschetz S. Stability by Lyapunov's Direct Method with Applications. Stability Theory by Lyapunov's Direct Method. Lakshmikantham V, Liu XZ. Stability Analysis in Terms of Two Measures.

## SIAM Journal on Control and Optimization

Singapore: World Scientific; Stability criteria in terms of two measures for functional differential equations. Applied Mathematics Letters.

Wang P, Lian H. On the stability in terms of two measures for perturbed impulsive integro-differential equations. Journal of Mathematical Analysis and Applications. Wang P, Zhan Z. Stability in terms of two measures of dynamic system on time scales. Computers and Mathematics with Applications.

Practical Stability of Nonlinear Systems. Zhang Y, Sun J. Practical stability of impulsive functional differential equations in terms of two measurements. Wang P, Liu X. Practical stability of impulsive hybrid differential systems in terms of two measures on time scales.

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Nonlinear Analysis: Theory, Methods and Applications. Practical stability in terms of two measures for discrete hybrid systems. Nonlinear Analysis: Hybrid Systems. Practical stability in terms of two measures for hybrid dynamic systems. Bulletin of the Polish Academy of Sciences.