Viscosity CBFs: Bridging the Control Barrier Function and Hamilton-Jacobi Reachability Frameworks in Safe Control Theory

TL;DR

The paper addresses bridging control barrier functions and Hamilton-Jacobi reachability to unify safe-control theory. It introduces viscosity CBFs, a non-differentiable generalization, and shows they correspond exactly to time-invariant CB-VFs, providing a Barrier Guarantee. It extends CB-VFs to nonlinear anti-discounting and proves key Hamilton-Jacobi PDE results to enable synthesis via max and limit operations. This bridge enables robust safety guarantees for non-smooth barrier functions and offers practical synthesis methods for safe controllers in complex dynamical systems.

Abstract

Control barrier functions (CBFs) and Hamilton-Jacobi reachability (HJR) are central frameworks in safe control. Traditionally, these frameworks have been viewed as distinct, with the former focusing on optimally safe controller design and the latter providing sufficient conditions for safety. A previous work introduced the notion of a control barrier value function (CB-VF), which is defined similarly to the other value functions studied in HJR but has certain CBF-like properties. In this work, we proceed the other direction by generalizing CBFs to non-differentiable ``viscosity'' CBFs. We show the deep connection between viscosity CBFs and CB-VFs, bridging the CBF and HJR frameworks. Through this bridge, we characterize the viscosity CBFs as precisely those functions which provide CBF-like safety guarantees (control invariance and smooth approach to the boundary). We then further show nice theoretical properties of viscosity CBFs, including their desirable closure under maximum and limit operations. In the process, we also extend CB-VFs to non-exponential anti-discounting and update the corresponding theory for CB-VFs along these lines.

Figures (2)

• Figure 1: Graphical abstract: in this work, we bridge Hamilton-Jacobi reachability (HJR) and control barrier function (CBFs). To do so, we extend the work on control barrier value functions (CB-VFs) in Choi-Herbert-CDC-CB-VFs-2021 and also introduce the notion of a viscosity CBF. A CB-VF is defined similarly to an avoid value function (a value function used in HJR for obstacle-avoidance tasks), except that it is anti-discounted according to a class $\mathcal{K}$ function $\alpha$. A viscosity CBF is similar to a standard CBF, except that the usual CBF inequality is only required to hold in a weak ("viscosity") sense. We show that a viscosity CBF is equivalent to a time-invariant CB-VF. We then show the set of viscosity CBFs is precisely the set of continuous functions which provide the Barrier Guarantee, i.e. certify control invariance of a set and bounds on the speed at which the system can approach the set boundary.
• Figure 2: Cartoon showing the level sets of an example immediate safety function $g:\mathbb{R}^2 \to \mathbb{R}$. Given a locally Lipschitz class $\mathcal{K}$ function $\alpha$, a time horizon $T$, and an initial state $x$ for which $g(x) > 0$, the value $v(x,T)$ of the corresponding CB-VF is equal to the initial immediate safety $g(x)$ when the control can keep the immediate safety value $g\left(\mathbf{x}_x^\mathbf{u}(t)\right)$ along the trajectory above or arbitrarily close to $\beta_\alpha(g(x),t)$ at all times $t \in [0,T]$. Otherwise, the CB-VF value is strictly less than the initial immediate safety.

Theorems & Definitions (42)

• Definition 1: Class $\mathcal{K}$; Definition 4.2 in Chapter 4.4 of Khalil-Nonlinear-Systems
• Definition 2: Class $\mathcal{K}\mathcal{L}$; Definition 4.3 in Chapter 4.4 of Khalil-Nonlinear-Systems
• Definition 3: Control Invariant Set
• Definition 4: Control Barrier Function
• Remark 1
• Definition 5: Barrier Guarantee
• Remark 2
• Lemma 1
• Definition 6: Viscosity CBF
• Example 1