Stability Analysis and Boundary Control of Coupled ODE-PDE Systems with a Class of Nonlinearity

Abstract

The research effort focusses on formulating boundary control for systems that have coupled ordinary differential equations (ODEs) and partial differential equations (PDEs), incorporating nonlinearity as characterized by the Wiener and Hammerstein models. The system coupling is considered through either the states or via input-output cascade configurations. The main objectives of the project include studying the formation and representation of ODE-PDE systems with nonlinearities described by both the Wiener and Hammerstein models. Additionally, the project aims to develop controllers and conduct stability analysis for these systems.

The key contributions of the project are the development of a backstepping controller for the Wiener ODE system coupled with a heat PDE system, along with performing stability analysis for the same. Furthermore, the research led to the design of a backstepping controller for a Hammerstein ODE system cascaded with a wave PDE system and the corresponding stability analysis. A similar contribution was made for the Hammerstein ODE system cascaded with a heat PDE system, where both the controller design and stability analysis were achieved. A significant future milestone is to extend the development to handle coupled ODE-PDE systems with different classes of nonlinearity, which is expected to enrich the real-world applications.

In summary, this research makes valuable contributions to boundary control strategies for nonlinear coupled ODE-PDE systems, providing initial steps for further work on more complex systems.