State and output feedback boundary control of time fractional PDE–fractional ODE cascades with space‐dependent diffusivity

Abstract

This study considers the boundary control problem of a time fractional partial differential equation (PDE) – fractional ordinary differential equation (ODE) cascaded system with spatially varying diffusivity and Dirichlet connection. Using the backstepping transformation, a full-state feedback law is obtained. Meanwhile, the well-posedness of kernel functions in this transformation is established theoretically. Subsequently, a Mittag-Leffler convergent boundary observer is derived, which composes with the proposed state feedback law to enable stabilisation by output feedback. With the designed state and output feedback controllers, the Mittag-Leffler stability of the closed-loop system is then proved by the fractional Lyapunov method. Finally, the results of numerical simulations are provided for fractional cascaded plants when neither kernel ODE nor kernel PDE has an explicit solution.