Estimator-Based Time-Varying Feedback Control for Uncertain, Anti-Stable Heat Equation in a Prescribed Finite Time

This paper studies prescribed-time (PT) stabilization for a heat equation with an unstable term at the uncontrolled boundary, subject to external disturbances and internal unknown mode uncertainties at the controlled boundary. A boundary time-varying output feedback control scheme based on disturbance estimation is developed, where the convergence time is independent of the initial condition and can be specified a priori. A disturbance estimator using boundary measurements and a time-varying tuning function enables prescribed-time estimation of both the disturbance and the system state. By integrating active disturbance rejection control with the backstepping technique, a boundary output feedback controller is derived. A simulation example from the burning process of a solid propellant rocket demonstrates the effectiveness of the proposed approach.