Input-to-state stability of an ODE-heat cascade system with disturbances

In this study, the authors consider the input-to-state stability of an ordinary differential equation (ODE)–heat cascade system with Dirichlet interconnection where the boundary control input is located at the right end of the heat equation and the disturbance is appeared as a non-homogeneous term in the ODE. Based on two backstepping transformations, they design a state feedback control law that guarantees the input-to-state stability of the closed-loop system. The well-posedness of the closed-loop system is presented by using the semi-group approach. Moreover, they design an output feedback control law by constructing an exponentially convergent observer. With the output feedback control, the input-to-state stability of the resulting closed-loop system is proven.