Fuzzy differential inclusion

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

$$ x'(t)  \in [ f(t, x(t))]^\alpha $$ 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
 * Artificial intelligence, Neural network,
 * Medical imaging
 * Robotics
 * Atmospheric dispersion modeling
 * Weather forecasting
 * Cyclone
 * Pattern recognition
 * Population biology