Boundary state and output feedback stabilisation of a coupled time fractional hyperbolic equation

In this paper, the boundary stabilisation of a coupled time fractional hyperbolic equation system is investigated by both state and output feedback. A target system is first introduced, which, by the semigroup method and the Lyapunov one, is proved to admit a unique solution and is Mittag-Leffler stable. A boundary state feedback controller and state observer are designed via the backstepping technique, with explicit formulae for the gain functions. An observer-based output feedback stabilising controller is then proposed for the controlled system. Some rigorous theoretical proofs are given to demonstrate that the resulting closed-loop system under the state and output feedback controllers has a unique solution and is Mittag-Leffler stable. Some numerical simulations are performed to validate the effectiveness of the proposed control approach.