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

In 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.