Stability results for the continuity equation

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 p L state norms with 1 p  , including the case p   . 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.

Event-Triggered Control of a Continuum Model of Highly Re-Entrant Manufacturing System

“…(ρ(x)) > 0 . The use of the logarithmic norm of the state is common in systems with positivity constraints (see [36,37]).…”

“…These continuum models describe the time evolution of the flow of manufactured products using the spatial distribution of product density as a key variable. Several contributions considering the control of boundary influx of parts with PI controller [48], Lyapunov-based design [43,44,45], small gain design [36], predictor-feedback design [49] or optimal [41,42]…”