Disturbance-to-state stabilization by output feedback of nonlinear ODE cascaded with a reaction–diffusion equation

In this paper, we consider the problem of disturbance-to-state stabilization for a class of nonlinear systems formed by a finite-dimensional ODE coupled with a one-dimensional heat diffusion PDE. The system is subject to both in-domain and boundary disturbances, and only partial state measurements are available. The ODE subsystem contains a nonlinear additive term with upper triangular structure, and the coupling between the two subsystems occurs through the boundary. We design an output feedback controller based on a high-gain observer combined with a backstepping transformation. The closed-loop system is formulated as a \(C_0\) $C_0$ -semigroup on a Hilbert space, and well-posedness is established using the Bari theorem. We construct Lyapunov functionals and derive explicit conditions under which the closed-loop system is globally exponentially input-to-state stable with respect to all external disturbances, thereby achieving disturbance-to-state stabilization.