Exponential input-to-state stabilization of an ODE cascaded with a reaction–diffusion equation subject to disturbances

This paper addresses the exponential input-to-state stabilization of an ODE cascaded with a reaction–diffusion equation subject to disturbances appearing in all channels. By applying the backstepping method and the sliding mode control approach, the discontinuous boundary feedback controller is designed to reject the matched disturbance and achieve the exponential input-to-state stability of the closed-loop system. The existence of generalized solution to the closed-loop system is obtained using the Galerkin approximation scheme. The effectiveness of the proposed controller is illustrated by some numerical simulations.