Continuous-domain theoretical analysis of the CBF safety filter

Establish the theoretical properties of the proposed control barrier function-based safety filter for the LWR traffic model in the continuous spatial domain.

Background

The paper develops and analyzes a discretized control barrier function (CBF) formulation for in-domain congestion prevention in the Lighthill–Whitham–Richards (LWR) traffic model. After spatial discretization, the relevant operator becomes a lower bidiagonal matrix, and the proposed primal-dual active set algorithm is shown to have convergence properties using the structure of an associated M-matrix.

The authors note that the corresponding continuous-space operator has analogous positivity properties. However, the paper does not establish the continuous-domain counterpart of the discrete theoretical analysis, leaving open the rigorous investigation of the CBF safety filter and its associated operator in the continuous spatial setting.

References

Indeed, the operator $\mathcal{L}\mathcal{L}*+\mathcal{I}_d/\beta$ is an elliptic operator, where $\mathcal{I}_d$ is the identity operator.The operator shares similar positivity properties with the matrix considered in the Appendix. We will investigate the theoretical analysis in continuous spatial domain in future work.

— Real-Time In-Domain Congestion Control for the LWR Traffic Model via Control Barrier Functions  (2608.13841 - Zhu et al., 14 Aug 2026) in Remark following Corollary 1, Section 3.3 (Controller Design Algorithm)