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Converse Theorems for Certificates of Safety and Stability

Pol Mestres, Jorge Cortés

TL;DR

This work tackles the existence and structure of certificates for safety and stability in nonlinear control. It develops converse theorems showing when control barrier functions ($CBF$) exist for a given safe set and introduces extended barrier functions ($eCBF$) to handle unbounded sets, ensuring safety certification under broader conditions. It then shifts to joint safety and stability, proving conditions for the existence of a control Lyapunov–barrier function ($CLBF$), strictly compatible CLF–CBF pairs, and compatible CLF–eCBF pairs, while identifying topological obstructions that limit these existences. A key takeaway is that while a $CBF$ or a $CLF$ may exist independently, ensuring a strictly compatible pair or a $CLBF$ can require additional hypotheses such as compactness or specific controller feasibility, and the paper provides both constructive results and counterexamples illustrating the limits. Overall, the extended framework and converse results deepen understanding of when safety and stability can be achieved together and guide controller design for safety-critical systems.

Abstract

Motivated by the key role of control barrier functions (CBFs) in assessing safety and enabling the synthesis of safe controllers in nonlinear control systems, this paper presents a suite of converse results on CBFs. Given any safe set, we first identify a set of general sufficient conditions which guarantee the existence of a CBF. Our technical analysis also enables us to define an extended notion of CBF which is always guaranteed to exist if the set is safe. We next turn our attention to the problem of joint safety and stability, and give conditions under which the notions of control Lyapunov-barrier function (CLBF) and compatible control Lyapunov function (CLF) and CBF pair are guaranteed to exist. Finally, we identify conditions under which a CLBF and a compatible CLF-CBF pair can be constructed from a non-compatible CLF-CBF pair. Throughout the paper, we intersperse different examples and counterexamples to motivate our results and position them within the state of the art.

Converse Theorems for Certificates of Safety and Stability

TL;DR

This work tackles the existence and structure of certificates for safety and stability in nonlinear control. It develops converse theorems showing when control barrier functions () exist for a given safe set and introduces extended barrier functions () to handle unbounded sets, ensuring safety certification under broader conditions. It then shifts to joint safety and stability, proving conditions for the existence of a control Lyapunov–barrier function (), strictly compatible CLF–CBF pairs, and compatible CLF–eCBF pairs, while identifying topological obstructions that limit these existences. A key takeaway is that while a or a may exist independently, ensuring a strictly compatible pair or a can require additional hypotheses such as compactness or specific controller feasibility, and the paper provides both constructive results and counterexamples illustrating the limits. Overall, the extended framework and converse results deepen understanding of when safety and stability can be achieved together and guide controller design for safety-critical systems.

Abstract

Motivated by the key role of control barrier functions (CBFs) in assessing safety and enabling the synthesis of safe controllers in nonlinear control systems, this paper presents a suite of converse results on CBFs. Given any safe set, we first identify a set of general sufficient conditions which guarantee the existence of a CBF. Our technical analysis also enables us to define an extended notion of CBF which is always guaranteed to exist if the set is safe. We next turn our attention to the problem of joint safety and stability, and give conditions under which the notions of control Lyapunov-barrier function (CLBF) and compatible control Lyapunov function (CLF) and CBF pair are guaranteed to exist. Finally, we identify conditions under which a CLBF and a compatible CLF-CBF pair can be constructed from a non-compatible CLF-CBF pair. Throughout the paper, we intersperse different examples and counterexamples to motivate our results and position them within the state of the art.
Paper Structure (11 sections, 10 theorems, 45 equations, 2 figures, 1 table)

This paper contains 11 sections, 10 theorems, 45 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Control Lyapunov and Barrier Functions
  5. Control Lyapunov-Barrier Functions
  6. Problem Statement
  7. Converse Results for Safety
  8. Converse Theorem for CBFs
  9. Extended Control Barrier Functions
  10. Converse Theorems for Joint Safety and Stability
  11. Conclusions

Key Result

Theorem 2.3

(CBFs certify safety ADA-SC-ME-GN-KS-PT:19): Let $\mathcal{C}\subset\mathbb{R}^{n}$, $h$ be a CBF of $\mathcal{C}$ for the system eq:control-sys, and $0$ be a regular value of $h$. Any Lipschitz controller $u_{\text{sf}}:\mathbb{R}^{n}\to\mathbb{R}^{m}$ that satisfies for all $x\in\mathcal{C}$ renders the set $\mathcal{C}$ forward invariant.

Figures (2)

  • Figure 1: Illustration of the set of points $(x,u)$ satisfying \ref{['eq:auxx']}. For every value of $x$, there are controls (colored in orange) that satisfy \ref{['eq:auxx']}.
  • Figure 2: Illustration of the different sets defined in the proof of Theorem \ref{['thm:converse-safe-stab']}\ref{['it:converse-safe-stab-clbf']}. The set $\mathcal{C}$ is the union of the orange, dark purple and light purple regions. The set $\mathcal{T}$ is the union of the orange and dark purple regions, and the set $\Pi$ is the union of the dark and light purple regions.

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Example 4.1
  • Remark 4.2
  • ...and 30 more