Are Control Density Functions Practical?

Overview

Safety-critical control typically seeks to avoid unsafe regions through Control Barrier Functions (CBFs), which enforce safety through gradient-based condition. Control Density Functions (CDFs) offer an alternative formulation that reasons about divergences rather than gradients.

This work summarizes the first semester of my PhD, where I investigated whether CDFs are more or less practical than CBFs.

Findings

• The additional divergence term has an ambiguous sign, which complicates safety verification; CDF certificates are easier to construct for some systems and harder for others.
• For controller synthesis, the divergence condition introduces derivatives of \(u\), preventing a direct QP formulation. A practical workaround reduces CDF synthesis to the same conditions as a CBF.
• Density functions may offer advantages for safety verification in specific problem classes, but are unlikely to simplify controller synthesis.

Citation

@inproceedings{pauker2026cdf,
  title   = {Are Control Density Functions Practical?},
  author  = {Pauker, Grant and Tu, Stephen and Lindemann, Lars},
  booktitle = {Learning for Dynamics and Control (L4DC)},
  year    = {2026}
}