Fuzzy differential inclusion explained

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 ^{\alpha} _{0 ]}

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 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

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

Notes and References

1. Book: Theory of Fuzzy Differential Equations and Inclusions . 978-0-367-39532-2 . Lakshmikantham . V. . Mohapatra . Ram N. . 11 September 2019 .
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 . 10.2298/FIL1506231M . 24898205 . 0354-5180. free .
3. Web site: Fuzzy differential inclusion in atmospheric and medical cybernetics .
4. Book: Tafazoli . Sina . Menhaj . Mohammad Bagher . 2009 IEEE Symposium on Computational Intelligence in Control and Automation . Fuzzy differential inclusion in neural modeling . March 2009 . https://ieeexplore.ieee.org/document/4982785 . 70–77 . 10.1109/CICA.2009.4982785. 978-1-4244-2752-9 . 5618541 .
5. Book: Min . Chao . Zhong . Yihua . Yang . Yan . Liu . Zhibin . 2012 9th International Conference on Fuzzy Systems and Knowledge Discovery . On the implicit fuzzy differential inclusions . May 2012 . https://ieeexplore.ieee.org/document/6234283 . 117–119 . 10.1109/FSKD.2012.6234283. 978-1-4673-0024-7 . 1952893 .
6. 10.1007/BF02228945 . 24898205 . Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology . 1992 . Antonelli . Peter L. . Křivan . Vlastimil . Open Systems & Information Dynamics . 1 . 2 . 217–232 . 123026730 .