Prescribed-time stabilization of semi-linear parabolic equations subject to distributed disturbance

In this paper, we investigate the problem of prescribed-time stabilization for a class of semilinear parabolic systems subject to spatiotemporal-varying disturbance via distributed control. By employing the time-varying feedback gain and disturbance suppression method, the proposed control law is continuous and stabilizes the closed-loop system within the prescribed time, where the convergence time is independent of initial values and can be given in advance as needed. When the upper bound of disturbance is known, we use a hyperbolic tangent function to restrain disturbance. While the upper bound of disturbance is unknown, we design the prescribed-time adaptive law and a prescribed-time disturbance observer estimating the disturbance itself. Some numerical examples are provided to verify the theoretical results.