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

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 openloop 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.

ISS-Lyapunov strictification via observer design and integral action control for a Korteweg-de-Vries equation

“…Indeed, such a methodology has been proved to be efficient for nonlinear systems. Among these techniques, we aim at using the forwarding method that has been first designed for finite-dimensional systems in cascade form [35,2] and then extended for some hyperbolic systems [47] in the regulation context, and for abstract systems [29] in the stabilization context. In [47], it is shown that a strict Lyapunov functional 1 is needed for open-loop stable systems that we aim at regulating.…”

“…We aim at showing that such a problem can be solved by means of an integral action and an output feedback control law. The proposed design is based on the forwarding method (see e.g., [47] or [29]). Note that in Section 5.1, we focus on the linearized version of the system.…”