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Boundary stabilization for time-space fractional diffusion-wave equation

In this paper, we investigate the boundary stabilization of an unstable time-space fractional diffusion-wave equation with Caputo time fractional derivative and spacial fractional Laplacian operator. Due to the spacial nonlocal operator leading to the difficulty in calculating spacial fractional derivative of Volterra integral transformation, the classic backstepping method fails to design the boundary controller. To overcome this difficulty, we are base on the state space decomposition so that the system is decomposed into infinite dimensional stable part and finite-dimensional unstable part, then we design a boundary feedback control for the finite dimensional unstable part, which is shown to actually stabilize the entire system.