Backstepping method for Stabilizing Fuzzy parabolic equation using Difference Method

In this work, a set of laws has been proposed as a group of feedback control stabilizing a boundary law for a linear fuzzy reaction-advection-diffusion equation. Stabilization is achieved by designing coordinate transformations that form recursive relationships; by using the fuzzy finite difference method, we can convert coordinates into other coordinates. This design process is unlimited to any specific kinds of boundary actuation and can handle systems with an arbitrarily finite number of eigenvalues for the unstable open-loop system. We noticed that there is another problem when converting coordinates, which is that the equation includes lower and upper functions, so we wrote the equations in the form of matrices and then converted them into ordinary differential equations. The problem of feedback, which becomes increasingly unbounded as the grid gets infinitely fine, is solved by carefully selecting the target system to which the original system is transformed. Then we stabilize the closed loop system and the regularity of control and solutions to the fuzzy reaction-advection-diffusion equation.