Mittag-Leffler Stabilization of an Unstable Time Fractional Hyperbolic PDE

This paper aims to study the Mittag-Leffler stabilization of an unstable time fractional hyperbolic partial differential equation by boundary control and boundary measurement. The backstepping method, the fractional Lyapunov method, and the semigroup theory are adopted in the investigation. A novel state feedback control via the Dirichlet boundary is designed to stabilize the controlled system. Based on the output signal, we first construct an observer that can recover the state of the original system, and then, we propose an observer-based stabilizing control law, under which the closed-loop system is shown to admit a unique solution and to be Mittag-Leffler stable. Finally, a benchmark example is presented to test the proposed theory.