Input-to-state stabilization of coupled parabolic PDEs subject to external disturbances

In this paper, we are concerned with the input-to-state stabilization of coupled parabolic partial differential equations, which is suffering the disturbances in all channels. By using the sliding mode control integrated with the backstepping approach, we design two boundary feedback controllers to reject the matched boundary disturbances, stabilize the whole coupled system in the absence of unmatched disturbances and achieve the input-to-state stability property of the closed-loop system with respect to the unmatched disturbances. We show the existence of the generalized solutions of the closed-loop system in the Hilbert energy space. Some numerical simulations are presented to illustrate the effectiveness of the proposed controllers.