Stabilizability and Solvability of Fuzzy Reaction-Diffusion Equation using Modified Backstepping Control Method for Matrix Differential Equation
DOI:
https://doi.org/10.11113/mjfas.v21n3.4249Keywords:
Fuzzy differential equation, Hukuhara derivative, fuzzy backstepping method, matrix differential equation, fuzzy reaction-diffusion equation.Abstract
In this article, an important type of fuzzy parabolic differential equations will be discussed, which is the one-dimensional fuzzy reaction-diffusion equation with fuzzy boundary conditions. This equation is one of the most widespread chemical fuzzy reaction-diffusion equations, as well as, studying the possibility of controlling and reducing the chemical pollution occurred in the chemical reactions. In order to reduce chemical contamination in the reaction medium, we observed that investigating this equation's stability is essential. In order to achieve stability, the fuzzy backstepping approach is proposed, which transforms the unstable system into a stable system after controlling the boundary conditions. Therefore, two different cases of Hukuhara derivatives must be considered, which are important in the study of fuzzy differential equations. Two cases are considered depending on the comparison between the lower and upper variable solution time derivative. Also, the proposed backstepping approach is applied based on the interval analysis of α-level sets. For this purpose, and in order to avoid the difficulty of separating the upper and lower solutions, the resulting non-fuzzy or crisp differential equations are converted into matrix differential equations, and then Consequently, we are able to remove the residual terms that are responsible for the instability of the open-loop. Moreover, this backstepping transformation is continuously invertible. Thus, the inverse transformation is used to obtain stabilizing state feedback for the original partial differential equation.
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