Forwarding-Lyapunov design for the stabilization of coupled ODEs and exponentially stable PDEs

TL;DR

This paper introduces a forwarding-based Lyapunov functional approach to stabilize a cascade system combining an exponentially stable PDE and a marginally stable ODE, ensuring global exponential stability.

Contribution

It develops a novel forwarding method using Lyapunov functionals and Sylvester equations for stabilization of coupled infinite-dimensional and finite-dimensional systems.

Findings

The closed-loop system achieves global exponential stability.

The approach applies under classical output regulation assumptions.

The method effectively stabilizes the cascade system with PDE and ODE components.

Abstract

This paper is about the stabilization of a cascade system composed by an infinite-dimensional system, that we suppose to be exponentially stable, and an ordinary differential equation (ODE), that we suppose to be marginally stable. The system is controlled through the infinite-dimensional system. Such a structure is particularly useful when applying the internal model approach on infinite-dimensional systems. Our strategy relies on the forwarding method, which uses a Lyapunov functional and a Sylvester equation to build a feedback-law. Under some classical assumptions in the output regulation theory, we prove that the closed-loop system is globally exponentially stable.