Stability Results for the Continuity Equation

TL;DR

This paper analyzes the stability of the 1-D continuity equation, providing comprehensive estimates in various norms and demonstrating applicability to complex nonlinear manufacturing models under feedback control.

Contribution

It offers new stability estimates for the 1-D continuity equation considering boundary disturbances and velocity inputs, with implications for larger nonlinear systems.

Findings

Stability estimates in all Lp norms and sup norm.

Dependence of stability gain on velocity.

Applicability to nonlinear manufacturing models.

Abstract

We provide a thorough study of stability of the 1-D continuity equation, which models many physical conservation laws. In our system-theoretic perspective, the velocity is considered to be an input. An additional input appears in the boundary condition (boundary disturbance). Stability estimates are provided in all Lp state norms with p>1, as well as in the sup norm. However, in our Input-to-State Stability estimates, the gain and overshoot coefficients depend on the velocity. Moreover, the logarithmic norm of the state appears instead of the usual norm. The obtained results can be used in the stability analysis of larger models that contain the continuity equation. In particular, it is shown that the obtained results can be used in a straightforward way for the stability analysis of non-local, nonlinear manufacturing models under feedback control.