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Construction of Control Lyapunov-Barrier Functions from CLF-CBF Pairs

Bo Wang, Miroslav Krstic

arXiv:2609.11746v1eess.SYcs.SY

TL;DR

Safe stabilization must combine convergence to an equilibrium with remaining in a prescribed safe set, but separate CLF and CBF inequalities can conflict. This paper characterizes CLBF decrease for functions formed from a CLF and CBF using a joint-level-set scalar weight, then constructs such functions and identifies a properness obstruction. Nonlinear examples demonstrate safe stabilization where feedback based on the original CLF violates safety.

  • Problem

    Separate CLF and CBF inequalities may conflict, and an input satisfying both is not guaranteed, motivating a unified CLBF construction from a given CLF and CBF.

    Source: §I · p. 1

  • Method

    The paper characterizes CLBF decrease using a nonnegative scalar weight admissible on joint level sets of (V, h), then constructs F from admissible weights through a first-order PDE.

    Source: §I · p. 1, §VI · p. 8

  • Results

    Two nonlinear examples demonstrate safe stabilization in cases where feedback based on the original CLF violates the safety constraint.

    Source: §VI · p. 8

  • Takeaways & Limitations

    Admissibility alone does not ensure properness, and an integrability-based obstruction identifies weights that cannot generate the required growth near the safety boundary.

    Source: B. An Obstruction to Properness · pp. 3–4, §VI · p. 8

    An admissible weight may grow too slowly as h approaches zero to produce a proper function.

    Source: B. An Obstruction to Properness · pp. 3–4

Abstract

from arXiv · show

This paper studies the construction of control Lyapunov-barrier functions (CLBFs) from a given control Lyapunov function (CLF) $V$ and control barrier function (CBF) $h$. We consider functions of the form $W=F(V,h)$ that increase with the CLF value and do not increase with the barrier value, and show that the CLBF decrease condition is completely characterized by a nonnegative scalar weight that governs the relative contributions of the CLF and CBF. Because this weight depends only on $(V,h)$, the same value must satisfy the decrease condition at all states sharing the same CLF and CBF values, which leads to a joint-level-set admissibility condition. We then construct CLBFs from admissible weights through a first-order partial differential equation (PDE) and obtain explicit multiplicative and power-type families as special cases. We further identify an integrability obstruction showing that admissibility alone does not guarantee properness. Two nonlinear examples illustrate the constructions and demonstrate safe stabilization in cases where feedback based on the original CLF violates the safety constraint.

I. INTRODUCTION

The paper asks how to construct a CLBF from a given CLF and CBF, motivated by possible conflicts between their separate decrease and safety conditions. It characterizes the decrease condition through a scalar weight and gives constructive families, while examples demonstrate safe stabilization where CLF-based feedback violates safety.

I. INTRODUCTION

  • Separate CLF and CBF inequalities can conflict, and a control input satisfying both is not guaranteed.The paper studies a single CLBF decrease condition as an alternative to enforcing both inequalities simultaneously.Source: §I · p. 1
  • Prior CLBF work includes constructions from CLF–CBF pairs, converse results, compatibility-based patching, and maximal CLBF characterizations.Other approaches construct feedback enforcing CLF and CBF inequalities directly.Source: §I · p. 1
  • A scalar weight κ(v, s) completely determines the CLBF decrease condition, and its value must satisfy that condition at every state sharing the same CLF and CBF values.This yields an admissibility condition on joint level sets of (V, h).Source: §I · p. 1
  • The paper constructs F from admissible weights by a first-order PDE, yielding multiplicative families and a power-type family under compatibility and boundary-growth conditions.Admissibility alone does not guarantee properness; an integrability-based obstruction is identified.Source: §I · p. 1
  • Two nonlinear examples illustrate the constructions and demonstrate safe stabilization where feedback based on the original CLF violates the safety constraint.One construction is nonmultiplicative.Source: §I · p. 2

II. PRELIMINARIES AND PROBLEM FORMULATION

The paper formulates safe stabilization on the interior of a safe set and seeks a proper CLBF formed from a given CLF and CBF. Its CLBF definition combines positive definiteness, properness, and a strict decrease condition.

II. PRELIMINARIES AND PROBLEM FORMULATION

  • The system is control-affine with locally Lipschitz dynamics, and the connected interior safe set contains the origin.The CBF h defines the safe set through C, while V is a positive definite, proper CLF on D.Source: §II · p. 2
  • The paper uses a CLBF on the open safe domain whose properness accounts for approaching the safety boundary and, for unbounded interiors, escaping to infinity.Source: §II · p. 2
  • A CLBF is positive definite, proper on the interior of the safe set, and satisfies a strict decrease implication when its control derivative is zero.Properness means sublevel sets are compact subsets of the safe domain; nonincreasing W keeps trajectories in such a set and therefore safe.Source: §II · p. 2
  • The problem is to construct a continuously differentiable F(V, h) that is a CLBF on the interior of the safe set.The formulation asks when this form satisfies the decrease condition, what obstructs properness, and how to construct F.Source: §II · p. 2
  • The construction considers F that increases with the CLF value and does not increase with the barrier value; its decrease condition is determined by one nonnegative scalar weight.Source: §II · p. 2

A. The Decrease Condition

For W=F(V,h), the CLBF decrease condition is governed by a nonnegative scalar weight that sets the relative contributions of the CLF and CBF gradients. Since the weight depends only on (V,h), it must satisfy the decrease condition across every state sharing those values.

A. The Decrease Condition

  • When Fv>0, the nonnegative weight κ determines the relative contributions of the CLF and CBF gradients; κ=0 recovers CLF decrease, while κ>0 adds dependence on h.The other partial derivative supplies only positive scaling.Source: A. The Decrease Condition · p. 2
  • A weight is admissible only if it satisfies the decrease condition at every state on the joint level set sharing the same (V,h) values.The characterization is equivalent to the CLBF decrease condition when the stated assumptions hold.Source: A. The Decrease Condition · pp. 2–3
  • Admissibility can be checked directly in state space, and the admissible set A(v,s) need not be an interval.The paper gives a linear-system example where the admissible set is disconnected despite a quadratic CLF and linear CBF.Source: A. The Decrease Condition · p. 3

B. An Obstruction to Properness

Admissibility of a weight ensures the CLBF decrease condition, but it does not ensure properness. The obstruction is that a weight integrable toward the safety boundary cannot generate the required boundary growth on a level-set strip.

B. An Obstruction to Properness

  • Admissibility and properness impose different requirements: κ = 0 is always admissible, yet an admissible choice may grow too slowly as s → 0 to make W proper.Admissibility supplies the decrease condition, not the boundary growth needed for properness.Source: B. An Obstruction to Properness · p. 3
  • A continuous weight on a joint-level-set strip extending to the safety boundary makes any F satisfying the stated ratio condition yield a nonproper W if the weight is integrable in s there.The strip must include [v0, v1] × (0, s0] within the range of (V, h).Source: B. An Obstruction to Properness · p. 3
  • In the linear example, the admissible 1/√s-type selection is integrable at s = 0 and therefore cannot produce a proper CLBF, whereas the 1/s-type selection is not ruled out by the obstruction.The comparison applies on any stated strip [v0, v1] × (0, s0] in the range of (V, h).Source: B. An Obstruction to Properness · p. 4

A. Construction via Characteristics

The construction turns an admissible weight into F by solving a first-order PDE along characteristic curves. The weight sets the ratio of partial derivatives, while a reference-line function assigns values to the curves; positivity and properness still require separate verification.

A. Construction via Characteristics

  • For a regular admissible weight κ with well-defined characteristic curves reaching a reference line, the theorem constructs F through a first-order PDE and yields a CLBF when the decrease condition, positive definiteness, and properness requirements hold.The construction assumes the characteristic solutions exist between s and the fixed reference value s_b and remain in the domain.Source: A. Construction via Characteristics · p. 4
  • Along each characteristic F is constant, and κ ≥ 0 gives F_s = −κF_v ≤ 0; with −F_s/F_v = κ, admissibility supplies the CLBF decrease condition.Positive definiteness and properness of W = F(V, h) must still be checked separately.Source: A. Construction via Characteristics · p. 4
  • The weight κ determines the characteristic curves and the ratio −F_s/F_v, while ψ assigns values to those curves through the reference line s = s_b.Different ψ choices can produce different F functions for the same κ.Source: A. Construction via Characteristics · p. 4
  • Applying Sontag’s formula to the constructed W yields an L_gW-type feedback with W decreasing away from the origin, and proper sublevel sets keep trajectories safe when forward solutions are well defined.A complete inverse-optimal treatment is left for future work.Source: A. Construction via Characteristics · p. 5
  • In the continued linear example, the selected weight yields a CLBF after positive definiteness and properness are verified, including divergence as h approaches zero.The argument also uses the stated bound on V in terms of s = h.Source: A. Construction via Characteristics · p. 5

B. Closed-Form Constructions

The section derives explicit CLBF constructions from CLF–CBF pairs, including multiplicative and power-type forms. These constructions require separate checks of the conditions that ensure decrease and properness.

B. Closed-Form Constructions

  • Choosing ψ(r) = r gives a closed-form construction whenever its right-hand side is well defined.For γ(v) > 0, the characteristic equation can be integrated explicitly.Source: B. Closed-Form Constructions · p. 5
  • Under Corollary 1’s hypotheses, the multiplicative construction yields a CLBF on Int(C).Its proof establishes positive definiteness and the decrease condition; properness is also required by the corollary.Source: B. Closed-Form Constructions · p. 5
  • Properness follows by bounding W-sublevel sets with compact V-sublevel sets and showing they cannot approach ∂C.Together with the CLBF decrease condition, this establishes that W is a CLBF on Int(C).Source: B. Closed-Form Constructions · pp. 5–6
  • Under CLF–CBF compatibility and the proposition’s conditions, the choice µ(s) = c/αh(s) gives a power-type CLBF.The construction procedure also requires verifying (63) and (65) and choosing c > 0 to satisfy (66).Source: B. Closed-Form Constructions · p. 6

V. EXAMPLES OF CLBF CONSTRUCTION

Two nonlinear examples construct CLBF feedback that achieves safe stabilization where feedback based on the original CLF may violate safety. The examples illustrate multiplicative and nonmultiplicative constructions, including one using characteristics.

Examples overview

  • Across both examples, numerical trajectories illustrate safe stabilization by constructed CLBF feedback when feedback based on the original CLF may violate safety.The construction conditions are verified analytically; numerical integration is used only to illustrate closed-loop trajectories.Source: §V · p. 6

Example 3 (Multiplicative CLBF under Compatibility)

  • The transformed system has a global CLF and an associated CBF, and the pair is compatible throughout Int(C) \ {0}.The compatibility verification shows that a suitable input can satisfy both inequalities.Source: §V · pp. 6–7
  • The CLF feedback crosses the boundary z = 1 despite decreasing V, whereas the proper CLBF feedback preserves safety and stabilizes the origin.The comparison uses k > 1/4 for the CLF feedback; the CLBF feedback is given by (79).Source: §V · p. 7, Figure 1 · p. 7
  • For κ(v,s)=cv/s with c>0, Corollary 1 gives a CLBF family for every c, although no fixed c satisfies cV(x)<α(|x|) throughout Int(C)\{0}; thus Proposition 2’s sufficient decay condition is not necessary.Source: §V · p. 7

Example 4 (Construction via Characteristics)

  • The characteristic construction also yields a proper, positive-definite nonmultiplicative CLBF, with admissible weights that can depend on both V and h.The integral term diverges as h approaches zero while V grows at infinity; one illustrated weight depends on both variables.Source: §V · pp. 7–8
  • In Example 4, the CLF feedback drives the trajectory across the safety boundary x1 = 1, whereas the CLBF feedback preserves Int(C) and steers the state toward the origin.Source: Figure 2 · p. 8

2. Hence, the CLBF feedback

For the Example 4 comparison, the chosen CLBF feedback preserves safety while CLF feedback crosses the boundary. The comparison uses W = V − ln h and identical controller gain and initial state.

Example 4

  • With l > 1/4, the CLF feedback decreases V but need not preserve safety with respect to C.Source: §2 · p. 8
  • The CLF feedback crosses x1 = 1, whereas the CLBF feedback keeps the trajectory in Int(C) and steers it toward the origin.The comparison uses W = V − ln h, l = 0.3, and x(0) = (0.9, 1.2).Source: §2 · p. 8
  • The numerical comparison uses W = V − ln h with q = 1, and both controllers use l = 0.3 from x(0) = (0.9, 1.2).Source: §2 · p. 8

VI. CONCLUSION

The paper characterizes CLBF decrease for functions of a CLF and CBF through a nonnegative weight admissible on their joint level sets. It constructs such functions from admissible weights and identifies an integrability obstruction to properness.

Conclusion

  • For W = F(V, h) with Fv > 0 and Fs ≤ 0, the decrease condition is characterized by a nonnegative weight admissible on joint level sets of (V, h).Source: §VI · p. 8
  • An admissible weight determines a first-order PDE construction of F, yielding general, multiplicative, and power-type families under suitable conditions.Source: §VI · p. 8
  • Admissibility alone does not ensure properness; an integrability obstruction applies to weights that cannot generate the required growth near the safety boundary.Source: §VI · p. 8
  • The examples demonstrate safe stabilization in cases where feedback based on the original CLF violates the safety constraint.Source: §VI · p. 8
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