Boundary stabilization for time-space fractional diffusion equation

This article is concerned with boundary stabilization for time-space fractional diffusion equation. The classic backstepping method is incapable of designing the boundary controller for such kind of problem due to the difficulty in calculating space fractional derivative of Volterra integral transformation. We overcome this difficulty by decomposing the system into two subsystems: one is a finite dimensional system that is unstable but is shown to be controllable, the other is an infinite dimensional system that is stable. By verifying the Kalman rank condition, we are able to propose a controller to stabilize the unstable finite dimensional system and prove that the proposed controller actually stabilizes the entire system.