Stabilization and uncertainty analysis of a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation

Abstract

AbstractIn this paper, we consider a time‐fractional reaction diffusion equation cascaded with a time‐fractional hyperbolic partial differential equation (PDE), where the time‐fractional reaction diffusion equation possesses space‐dependent diffusivity and the time‐fractional hyperbolic equation acts as a boundary source for the time‐fractional reaction diffusion equation. Then, the control input is imposed at the boundary of the time‐fractional hyperbolic PDE. Two problems are investigated, namely, the stabilization by state feedback and the stability robustness of the closed‐loop system against small perturbations in the diffusion and reaction coefficients. By the backstepping transformation, we propose a boundary controller and show that this controller solves the boundary stabilization problem using the fractional Lyapunov method. Robustness to small perturbations in diffusion and reaction coefficients is proved. Finally, a numerical example is provided to test the effectiveness of the proposed synthesis for the stabilization problem when the kernel equations have not an explicit solution.