ISS in Different Norms for 1-D Parabolic PDES With Boundary Disturbances

TL;DR

This paper establishes input-to-state stability (ISS) estimates for 1-D parabolic PDEs with boundary and distributed disturbances across various norms, using eigenfunction expansion and finite-difference schemes, with applications to thermoelasticity.

Contribution

It introduces a novel proof approach for ISS in PDEs without an ISS Lyapunov functional, refining maximum principles and applying results to thermoelastic models with nonlocal boundary conditions.

Findings

ISS estimates in multiple norms for 1-D parabolic PDEs

Refinement of the maximum principle for heat equations

Global exponential stability conditions for thermoelasticity models

Abstract

For 1-D parabolic PDEs with disturbances at both boundaries and distributed disturbances we provide ISS estimates in various norms. Due to the lack of an ISS Lyapunov functional for boundary disturbances, the proof methodology uses (i) an eigenfunction expansion of the solution, and (ii) a finite-difference scheme. The ISS estimate for the sup-norm leads to a refinement of the well-known maximum principle for the heat equation. Finally, the obtained results are applied to quasi-static thermoelasticity models that involve nonlocal boundary conditions. Small-gain conditions that guarantee the global exponential stability of the zero solution for such models are derived.