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]