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Eliminating Persistent Boundary Residence via Matrosov-Type Auxiliary Functions

Tianyu Han, Guangwei Wang, Bo Wang

Abstract

Control barrier functions enforce safety by guaranteeing forward invariance of an admissible set. Under standard (non-strict) barrier conditions, however, forward invariance alone does not prevent trajectories from remaining on the boundary of the safe set for arbitrarily long time intervals, potentially leading to boundary sticking or deadlock phenomena. This paper studies the elimination of persistent boundary residence under forward-invariant barrier conditions. Inspired by Matrosov-type arguments, we introduce an auxiliary function framework that preserves forward invariance while excluding infinite-time residence within boundary layers. Sufficient conditions are established under which any trajectory can only remain in a prescribed neighborhood of the boundary for finite time, thereby restoring boundary-level liveness without altering forward invariance. The proposed construction does not rely on singular barrier formulations or controller-specific modifications, and can be incorporated into standard safety-critical control architectures. Numerical examples illustrate the removal of boundary sticking behaviors while maintaining safety across representative systems.

Eliminating Persistent Boundary Residence via Matrosov-Type Auxiliary Functions

Abstract

Control barrier functions enforce safety by guaranteeing forward invariance of an admissible set. Under standard (non-strict) barrier conditions, however, forward invariance alone does not prevent trajectories from remaining on the boundary of the safe set for arbitrarily long time intervals, potentially leading to boundary sticking or deadlock phenomena. This paper studies the elimination of persistent boundary residence under forward-invariant barrier conditions. Inspired by Matrosov-type arguments, we introduce an auxiliary function framework that preserves forward invariance while excluding infinite-time residence within boundary layers. Sufficient conditions are established under which any trajectory can only remain in a prescribed neighborhood of the boundary for finite time, thereby restoring boundary-level liveness without altering forward invariance. The proposed construction does not rely on singular barrier formulations or controller-specific modifications, and can be incorporated into standard safety-critical control architectures. Numerical examples illustrate the removal of boundary sticking behaviors while maintaining safety across representative systems.
Paper Structure (8 sections, 4 theorems, 67 equations, 3 figures)

This paper contains 8 sections, 4 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and Problem Statement
  3. Main Results
  4. Design Examples
  5. Single-Integrator System
  6. Double-Integrator System
  7. Nonholonomic Unicycle
  8. Conclusion

Key Result

Theorem 3

Assume that there are constants $\bar{R}>\underline{R}>0$ and $L>0$, functions $\underline{\alpha},\bar{\alpha}\in\mathcal{K}$, and continuous functions for which $\dot{V}_1(t,x)$ and $\dot{V}_2(t,x)$ are continuous and the following hold: Then the origin of eq:tv-system is uniformly asymptotically stable.$\square$

Figures (3)

  • Figure 1: Comparison of trajectory behaviors for the planar single-integrator system. (a) Standard CLF-CBF-QP controller exhibiting undesired asymptotically stable equilibria on the safety boundary. (b) Proposed controller eliminating boundary convergence.
  • Figure 2: Trajectory behaviors for the planar double-integrator system with the proposed controller. (a) Static obstacle, (b) Moving obstacle.
  • Figure 3: Comparison of trajectory behaviors for the unicycle system. (a) Standard CBF-QP filter \ref{['eq:unicycle16']} exhibiting boundary sticking. (b) Proposed controller \ref{['eq:unicycle19']} eliminating boundary sticking.

Theorems & Definitions (9)

  • Definition 1: Safety
  • Definition 2: Barrier Function
  • Theorem 3: Matrosov theorem ROUHABLAL
  • Theorem 4: Exclusion of Persistent Boundary Residence
  • proof
  • Lemma A.1
  • proof
  • Lemma A.2
  • proof