Small-Gain Stability Analysis of Hyperbolic-Parabolic PDE Loops

TL;DR

This paper establishes small-gain based stability conditions for hyperbolic-parabolic PDE loops in one dimension, addressing challenges in chemical underground movement modeling and hyperbolic PDE control, with results on ISS and exponential stability.

Contribution

It introduces novel small-gain stability analysis techniques for hyperbolic-parabolic PDE interconnections, including boundary disturbances and delay effects, with generalizable results.

Findings

ISS conditions for boundary disturbances in parabolic-hyperbolic systems

Delay-independent exponential stability criteria

Application to chemical movement and wave damping models

Abstract

This work provides stability results in the spatial sup norm for hyperbolic-parabolic loops in one spatial dimension. The results are obtained by an application of the small-gain stability analysis. Two particular cases are selected for the study because they contain challenges typical of more general systems (to which the results are easily generalizable but at the expense of less pedagogical clarity and more notational clutter): (i) the feedback interconnection of a parabolic PDE with a first-order zero-speed hyperbolic PDE with boundary disturbances, and (ii) the feedback interconnection, by means of a combination of boundary and in-domain terms, of a parabolic PDE with a first-order hyperbolic PDE. The first case arises in the study of the movement of chemicals underground and includes the wave equation with Kelvin-Voigt damping as a subcase. The second case arises when we apply backstepping to a pair of hyperbolic PDEs that is obtained by ignoring diffusion phenomena. Moreover, the second case arises in the study of parabolic PDEs with distributed delays. In the first case, we provide sufficient conditions for ISS in the spatial sup norm with respect to boundary disturbances. In the second case, we provide (delay-independent) sufficient conditions for exponential stability in the spatial sup norm.