Fuzzy differential inclusion

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

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

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

Second order differential

The second order differential is

x ″ ( t ) ∈ [ k x ] α {\displaystyle x''(t)\in [kx]^{\alpha }} where k ∈ [ K ] α {\displaystyle k\in [K]^{\alpha }} , K {\displaystyle K} is trapezoidal fuzzy number ( − 1 , − 1 / 2 , 0 , 1 / 2 ) {\displaystyle (-1,-1/2,0,1/2)} , and x 0 {\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.