Boundary state and output feedbacks for underactuated systems of coupled time-fractional PDEs with different space-dependent diffusivity

Abstract

This paper considers the boundary stabilisation problem of an underactuated system of coupled time-fractional partial differential equations (PDEs) with different space-dependent diffusivity by state and observer-based output feedback. For this underactuated system, there are n fractional equations, and n−1 of n equations are actuated at the boundary inputs, i.e. n−1 available inputs control the entire system of n fractional equations. In other words, we do not need to control all equations. Using the backstepping mapping, the Dirichlet boundary controller is derived to enable the Mittag–Leffler stability of the closed-loop system with the aid of the Lyapunov direct method. The Mittag–Leffler convergence of designed boundary observer and output feedback stabilisation are subsequently proved. Finally, a fractional numerical example is provided to support the effectiveness of the proposed synthesis for the case when the kernel matrix PDE has not an explicit solution.