ISS With Respect to Boundary Disturbances for 1-D Parabolic PDES

TL;DR

This paper demonstrates that input-to-state stability (ISS) can be established for boundary disturbances in 1-D parabolic PDEs, challenging previous beliefs and providing new analysis methods for boundary control systems.

Contribution

It introduces a novel approach to prove ISS for boundary disturbances in parabolic PDEs without transforming boundary inputs, using time-varying state space analysis.

Findings

ISS with respect to boundary disturbances is achievable for parabolic PDEs.

The methodology allows comparison of gain coefficients in transport PDEs.

ISS with respect to control actuator errors is established for boundary feedback control.

Abstract

Due to unbounded input operators in partial differential equations (PDEs) with boundary inputs, there has been a long-held intuition that input-to-state stability (ISS) properties and finite gains cannot be established with respect to disturbances at the boundary. This intuition has been reinforced by many unsuccessful attempts, as well as by the success in establishing ISS only with respect to the derivative of the disturbance. Contrary to this intuition, we establish such a result for parabolic PDEs. Our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time-varying subsets of the state space. The obtained results are used for the comparison of the gain coefficients of transport PDEs with respect to inlet disturbances and for the establishment of the ISS property with respect to control actuator errors for parabolic systems under boundary feedback control.