Boundary Stabilization with restricted observability

TL;DR

This paper studies stabilization of one-dimensional hyperbolic balance laws with boundary controls when only restricted boundary observations are available. It develops a framework that transforms physical boundary conditions to Riemann coordinates and uses a rank-one boundary map $K = uv^T$ to obtain explicit stability criteria, including the condition $|v^T T| \cdot |((\mathcal{A}-\mathcal{K}^0)T)^{-1} u|<1$. The authors apply the results to density-flow and shallow-water models, providing both analytical conditions and numerical illustrations that show when limited observability preserves stabilization or may prevent it. The work offers practical guidance for designing boundary-feedback laws when only physical variables are observable, with potential extensions to quasilinear systems.

Abstract

Lyapunov functions are popularly used to investigate the stabilization problem of systems of hyperbolic conservation laws with boundary controls. In real life applications often not every boundary value can be observed. In this work, we show the stabilization under a restricted boundary observability. Thereby, we apply the boundary control directly on the observed (physical) variables. Using well-known stabilization results from the literature, we also discuss examples such as a density flow model or the Saint-Venant equations. This shows that a restricted observation can result in more restrictive control choices or can prevent the system from stabilizing.

Figures (2)

• Figure 1: Discrete $L^2-$norm on a logarithmic scale for $\mathcal{K}^0_{1,1}\in\{0.01,0,-0.01\}$ (left) and $\mathcal{K}^0_{1,1}\in\{-1,-100\}$ (right).
• Figure 2: Discrete $L^2-$norm on a logarithmic scale for the different parameter sets of Table \ref{['tab:para']}

Theorems & Definitions (14)

• Definition 2.1
• Theorem 2.2
• Corollary 1
• Definition 3.1
• Example 1
• Lemma 3.2
• proof
• Theorem 3.3
• Remark 1
• Example 2