Fuzzy differential inclusion

Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh.{{Cite book |title=Theory of Fuzzy Differential Equations and Inclusions |url=https://www.routledge.com/Theory-of-Fuzzy-Differential-Equations-and-Inclusions/Lakshmikantham-Mohapatra/p/book/9780367395322 | isbn=978-0-367-39532-2 |last1=Lakshmikantham |first1=V. |last2=Mohapatra |first2=Ram N. |date=11 September 2019 }}{{Cite journal |last1=Min |first1=Chao |last2=Liu |first2=Zhi-bin |last3=Zhang |first3=Lie-hui |last4=Huang |first4=Nan-jing |date=2015 |title=On a System of Fuzzy Differential Inclusions |url=https://www.jstor.org/stable/24898205 |journal=Filomat |volume=29 |issue=6 |pages=1231–1244 |doi=10.2298/FIL1506231M |jstor=24898205 |issn=0354-5180|doi-access=free }}

$x'(t) \in [ f(t , x(t))]^\alpha$

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

Cybernetics{{Cite web |title=Fuzzy differential inclusion in atmospheric and medical cybernetics |url=https://www.isibang.ac.in/~kaushik/kaushik_files/tumor.pdf}}
Artificial intelligence, Neural network,{{Cite book |last1=Tafazoli |first1=Sina |last2=Menhaj |first2=Mohammad Bagher |title=2009 IEEE Symposium on Computational Intelligence in Control and Automation |chapter=Fuzzy differential inclusion in neural modeling |date=March 2009 |chapter-url=https://ieeexplore.ieee.org/document/4982785 |pages=70–77 |doi=10.1109/CICA.2009.4982785|isbn=978-1-4244-2752-9 |s2cid=5618541 }}{{Cite book |last1=Min |first1=Chao |last2=Zhong |first2=Yihua |last3=Yang |first3=Yan |last4=Liu |first4=Zhibin |title=2012 9th International Conference on Fuzzy Systems and Knowledge Discovery |chapter=On the implicit fuzzy differential inclusions |date=May 2012 |chapter-url=https://ieeexplore.ieee.org/document/6234283 |pages=117–119 |doi=10.1109/FSKD.2012.6234283|isbn=978-1-4673-0024-7 |s2cid=1952893 }}
Medical imaging
Robotics
Atmospheric dispersion modeling
Weather forecasting
Cyclone
Pattern recognition
Population biology{{Cite journal |doi=10.1007/BF02228945 |jstor=24898205 |title=Fuzzy differential inclusions as substitutes for stochastic differential equations in population biology |year=1992 |last1=Antonelli |first1=Peter L. |last2=Křivan |first2=Vlastimil |journal=Open Systems & Information Dynamics |volume=1 |issue=2 |pages=217–232 |s2cid=123026730 }}

References

Category:Dynamical systems

Category:Variational analysis

Category:Fuzzy logic