Stabilizability of fuzzy heat equation based on fuzzy Lyapunov function

This work aims to prove the stability of the fuzzy heat equation with non-fuzzy boundary conditions. Also, for problems with uncertainty, there is a difficulty in studying and proving the stability of fuzzy differential equations, since here we cannot use the same methodologies used when proving the stability of nonfuzzy differential equations. Therefore, we consider two different cases of Hukuhara differentiability, which are given by interpreting the fuzzy functions and fuzzy numbers into crisp intervals through applying level sets with interval boundaries symbolized as the lower and upper crisp functions related to fuzzy solution. The direct Lyapunov method and Poincaré inequality are used for this purpose, as an application of the theory, in the first case of Hukuhara derivative and we are able to prove that the fuzzy heat equation is stable. However, in the second case, we are in contact with a difficulty of proving the stability using the Lyapunov function and the Lyapunov theorem due to the presence of the upper and lower functions in the same equation. Therefore, we resorted to other methods and give the ability to prove the stability of the fuzzy heat equation. Finally, an illustrative example is given as an enforcement for the sake of exhibiting and evaluating the stabilized fuzzy solution.