Fuzzy differential inclusion

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

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

Suppose ${\displaystyle f(t,x(t))}$ 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 ${\displaystyle \mathbb {R} ^{n}}$.

Second order differential

The second order differential is

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

Applications

Fuzzy differential inclusion (FDI) has applications in

• Cybernetics[3]
• Artificial intelligence, Neural network,[4][5]
• Medical imaging
• Robotics
• Atmospheric dispersion modeling
• Weather forecasting
• Cyclone
• Pattern recognition
• Population biology[6]

References

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.