Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE

TL;DR

This paper develops a forwarding-based feedback control method to stabilize coupled infinite-dimensional systems with nonlinear ODE components, ensuring global asymptotic stability under certain conditions.

Contribution

It extends the forwarding control technique to infinite-dimensional systems coupled with nonlinear ODEs, providing new stability conditions and well-posedness results.

Findings

Established sufficient conditions for stability and well-posedness.

Applied the method to a transport equation coupled with an ODE.

Demonstrated global asymptotic stability of the closed-loop system.

Abstract

This paper deals with the stabilization of a coupled system composed by an infinite-dimensional system and an ODE. Moreover, the control, which appears in the dynamics of the ODE, is subject to a general class of nonlinearities. Such a situation may arise, for instance, when the actuator admits a dynamics. The open-loop ODE is exponentially stable and the open-loop infinite-dimensional system is dissipative, i.e., the energy is nonincreasing, but its equilibrium point is not necessarily attractive. The feedback design is based on an extension of a finite-dimensional method, namely the forwarding method. We propose some sufficient conditions that imply the well-posedness and the global asymptotic stability of the closed-loop system. As illustration, we apply these results to a transport equation coupled with an ODE.