Fuzzy differential inclusion

Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh.^[1]^[2]

$x'(t)\in [f(t,x(t))]^{\alpha }$ with $x(0)\in [x_{0}]^{\alpha }$

Suppose $f(t,x(t))$ is a fuzzy valued continuous function on Euclidean space. Then it is the collection of all normal, upper semi-continuous, convex, compactly supported fuzzy subsets of $\mathbb {R} ^{n}$ .

Second order differential

The second order differential is

$x''(t)\in [kx]^{\alpha }$ where $k\in [K]^{\alpha }$ , $K$ is trapezoidal fuzzy number $(-1,-1/2,0,1/2)$ , and $x_{0}$ is a trianglular fuzzy number (-1,0,1).

Applications

Fuzzy differential inclusion (FDI) has applications in

References

^ Lakshmikantham, V.; Mohapatra, Ram N. (11 September 2019). Theory of Fuzzy Differential Equations and Inclusions. ISBN 978-0-367-39532-2.
^ Min, Chao; Liu, Zhi-bin; Zhang, Lie-hui; Huang, Nan-jing (2015). "On a System of Fuzzy Differential Inclusions". Filomat. 29 (6): 1231–1244. doi:10.2298/FIL1506231M. ISSN 0354-5180. JSTOR 24898205.
^ "Fuzzy differential inclusion in atmospheric and medical cybernetics" (PDF).
^ Tafazoli, Sina; Menhaj, Mohammad Bagher (March 2009). "Fuzzy differential inclusion in neural modeling". 2009 IEEE Symposium on Computational Intelligence in Control and Automation. pp. 70–77. doi:10.1109/CICA.2009.4982785. ISBN 978-1-4244-2752-9. S2CID 5618541.
^ Min, Chao; Zhong, Yihua; Yang, Yan; Liu, Zhibin (May 2012). "On the implicit fuzzy differential inclusions". 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery. pp. 117–119. doi:10.1109/FSKD.2012.6234283. ISBN 978-1-4673-0024-7. S2CID 1952893.
^ Antonelli, Peter L.; Křivan, Vlastimil (1992). "Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology". Open Systems & Information Dynamics. 1 (2): 217–232. doi:10.1007/BF02228945. JSTOR 24898205. S2CID 123026730.

[1] Lakshmikantham, V.; Mohapatra, Ram N. (11 September 2019). Theory of Fuzzy Differential Equations and Inclusions. ISBN 978-0-367-39532-2.

[2] Min, Chao; Liu, Zhi-bin; Zhang, Lie-hui; Huang, Nan-jing (2015). "On a System of Fuzzy Differential Inclusions". Filomat. 29 (6): 1231–1244. doi:10.2298/FIL1506231M. ISSN 0354-5180. JSTOR 24898205.

[3] "Fuzzy differential inclusion in atmospheric and medical cybernetics" (PDF).

[4] Tafazoli, Sina; Menhaj, Mohammad Bagher (March 2009). "Fuzzy differential inclusion in neural modeling". 2009 IEEE Symposium on Computational Intelligence in Control and Automation. pp. 70–77. doi:10.1109/CICA.2009.4982785. ISBN 978-1-4244-2752-9. S2CID 5618541.

[5] Min, Chao; Zhong, Yihua; Yang, Yan; Liu, Zhibin (May 2012). "On the implicit fuzzy differential inclusions". 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery. pp. 117–119. doi:10.1109/FSKD.2012.6234283. ISBN 978-1-4673-0024-7. S2CID 1952893.

[6] Antonelli, Peter L.; Křivan, Vlastimil (1992). "Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology". Open Systems & Information Dynamics. 1 (2): 217–232. doi:10.1007/BF02228945. JSTOR 24898205. S2CID 123026730.

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