Input-to-state stabilization of an ODE cascaded with a parabolic equation involving Dirichlet-Robin boundary disturbances

This paper focuses on the input-to-state stabilization problem for an ordinary differential equation (ODE) cascaded by parabolic partial differential equation (PDE) in the presence of Dirichlet-Robin boundary disturbances, as well as in-domain disturbances. For the cascaded system with a Dirichlet pointwise interconnection, the ODE takes the value of a Robin boundary condition at the ODE-PDE interface as its direct input, and the PDE is driven by a Dirichlet boundary input at the opposite end. We first employ the backstepping method to design a boundary controller and to decouple the cascaded system. This decoupling facilitates independent stability analysis of the PDE and ODE systems sequentially. Then, to address the challenges posed by Dirichlet boundary disturbances to the application of the classical Lyapunov method, we utilize the generalized Lyapunov method to establish the ISS in the max-norm for the cascaded system involving Dirichlet boundary disturbances and two other types of disturbances. The obtained result indicates that even in the presence of different types of disturbances, ISS analysis can still be conducted within the framework of Lyapunov stability theory. For the well-posedness of the target system, it is conducted by using the technique of lifting and the semigroup method. Finally, numerical simulations are conducted to illustrate the effectiveness of the proposed control scheme and ISS properties for a cascaded system with different disturbances.

💡 Research Summary

The paper addresses the input‑to‑state stabilization (ISS) problem for a cascade composed of a finite‑dimensional ordinary differential equation (ODE) and a one‑dimensional linear parabolic partial differential equation (PDE) when both Dirichlet‑Robin boundary disturbances and distributed (in‑domain) disturbances are present. The ODE receives the value of the PDE at the left boundary (a Robin‑type interface) as its input, while the PDE is actuated at the opposite end by a Dirichlet boundary control. The authors adopt a purely backstepping‑based design, avoiding sliding‑mode control and its associated chattering, to construct a continuous boundary feedback law.

First, a backstepping transformation is introduced: \