Input-to-State Stabilization for an ODE Cascaded by a Parabolic PIDE with Disturbances

This paper investigates the input-to-state stabilization problem of an ODE cascaded by a parabolic partial integro-differential equation (PIDE) subject to disturbances. First, the backstepping approach and the sliding mode control method are used to design the discontinuous boundary feedback control law. Then, a Galerkin approximation scheme is constructed to show the existence of solution to the closed-loop system. With the Lyapunov approach, the input-to-state stability of the closed-loop system is obtained. Finally, the simulation results are presented to illustrate the effectiveness of the proposed control law.