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

TL;DR

This paper develops a boundary control strategy using backstepping to achieve input-to-state stability for an ODE coupled with a parabolic PDE under boundary disturbances, with proofs and simulations confirming effectiveness.

Contribution

It introduces a novel backstepping-based boundary control method for cascaded ODE-PDE systems with Dirichlet-Robin disturbances, establishing ISS in the max-norm.

Findings

Successfully decouples the cascaded system for stability analysis.

Establishes ISS in the presence of boundary and in-domain disturbances.

Validates the approach with numerical simulations.

Abstract

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.