Fuzzy differential inclusion

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

$x^{\prime }(t)\in [f(t,x(t)){]}^{\alpha }$ 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 ${ℝ}^n$.

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
• Artificial intelligence, Neural network,
• Medical imaging
• Robotics
• Atmospheric dispersion modeling
• Weather forecasting
• Cyclone
• Pattern recognition
• Population biology