Recurrent Control Lyapunov Functions (R-CLFs)

• R-CLFs are defined by finite-time recurrence conditions that require the tracking error to contract at least once within every fixed horizon, replacing continuous monotonic decay.
• They integrate reduced-order CBFs with full-order model dynamics by constructing a recurrent barrier function that lifts safety guarantees to high-order systems.
• The synthesis method employs norm-based tracking functions and explicit error bounds, offering a computationally efficient alternative to SOS or HJ reachability programs.

Searching arXiv for the specified paper to ground the article in the cited source. Recurrent Control Lyapunov Functions, as developed through Recurrent Tracking Functions (RTFs), form a safety-critical control construction for high-order nonlinear systems in layered architectures where direct synthesis of valid Control Barrier Functions (CBFs) for the full-order model (FoM) is computationally intractable. The central idea is to design CBFs on a reduced-order model (RoM) while simultaneously regulating FoM dynamics through a tracking mechanism that does not require monotonic Lyapunov decrease at every instant. Instead, RTFs impose a weaker finite-time recurrence condition: the tracking measure must contract at least once within each horizon of length $\tau$. In "Safety-Critical Control via Recurrent Tracking Functions" (Liu et al., 1 Oct 2025), this relaxation is used to construct recurrent CBFs (RCBFs) whose zero-superlevel set is control $\tau$-recurrent, yielding safety guarantees for FoMs when the RTF conditions hold.

1. Formal definition of Recurrent Tracking Functions

The paper defines the closed-loop FoM trajectory as $x(t)$ under a layered control law, its projection onto the RoM as $z(t)=\Pi(x(t))$, and a safe reference trajectory generated by a RoM CBF as $z_s(t)$. The tracking error is

$$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$

A continuous function

$$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$

is a Recurrent Tracking Function over a compact set $S\subset\mathbb{R}^n\times\mathbb{R}^n$, with $0\in\mathrm{int}(S)$, if there exist constants $a_1,a_2,\beta,\tau>0$ such that for every $\tau$0 two conditions hold (Liu et al., 1 Oct 2025).

First, $\tau$1 satisfies positive definiteness and linear error bounds: $\tau$2

Second, $\tau$3 satisfies $\tau$4-exponential $\tau$5-recurrence: $\tau$6 where

$\tau$7

The paper states that condition (15) replaces the usual $\tau$8 requirement by demanding that, at least once every $\tau$9 seconds, $x(t)$0 has shrunk by a factor $x(t)$1 (Liu et al., 1 Oct 2025). This makes the construction explicitly non-monotone: transient excursions of the tracking error are permitted so long as recurrent contraction occurs.

2. Relation to classical Control Lyapunov Functions

The conceptual contrast with a classical Control Lyapunov Function (CLF) is one of the defining features of the framework. A classical CLF requires a continuously differentiable $x(t)$2 satisfying

$x(t)$3

for some $x(t)$4 at every instant (Liu et al., 1 Oct 2025).

An RTF instead requires only the finite-time recurrence inequality

$x(t)$5

so exponential contraction must occur at least once within every horizon $x(t)$6, not continuously. The paper characterizes this as strictly weaker than the classical pointwise decay condition while still guaranteeing exponential convergence of $x(t)$7 through Theorem 2 (Liu et al., 1 Oct 2025).

This weaker requirement is significant because systematic synthesis methods for monotone Lyapunov tracking functions are stated to exist only for fully-actuated systems, whereas the RTF relaxation is introduced precisely to overcome that limitation in layered safety-critical control of high-order nonlinear systems. A plausible implication is that the RTF formalism enlarges the class of error dynamics for which verifiable tracking certificates can be constructed without requiring instantaneous dissipation.

3. Recurrent CBF construction and safety theorems

The paper’s main theoretical bridge from RoM safety to FoM safety is the construction of a recurrent CBF from a RoM CBF and an RTF. The starting point is a RoM barrier function $x(t)$8 satisfying the linear-class-$x(t)$9 CBF condition

$z(t)=\Pi(x(t))$0

Given an RTF $z(t)=\Pi(x(t))$1 over $z(t)=\Pi(x(t))$2 with rate $z(t)=\Pi(x(t))$3 and recurrence $z(t)=\Pi(x(t))$4, Theorem 3 defines

$z(t)=\Pi(x(t))$5

where $z(t)=\Pi(x(t))$6 bounds $z(t)=\Pi(x(t))$7 and $z(t)=\Pi(x(t))$8 is the overshoot constant in the tracking bound (Liu et al., 1 Oct 2025).

Under these assumptions, the paper establishes two conclusions. First, the set

$z(t)=\Pi(x(t))$9

is a control $z_s(t)$0-recurrent set. Second, $z_s(t)$1 is a Recurrent Control Barrier Function on $z_s(t)$2 (Liu et al., 1 Oct 2025).

Theorem 4 then gives the safety assessment result: if the initial condition satisfies $z_s(t)$3, then under any control law satisfying the RTF condition (15), the full-order trajectory never violates the original RoM safety constraint $z_s(t)$4. Equivalently, $z_s(t)$5 remains in the safe FoM set for all $z_s(t)$6 (Liu et al., 1 Oct 2025).

These theorems provide the article’s core notion of an R-CLF-like object in context: the RTF plays the role traditionally associated with a tracking Lyapunov function, but the resulting safety certificate is a recurrent barrier rather than a standard barrier predicated on pointwise monotone decay.

4. Synthesis methodology

A central result in the paper is a converse-style synthesis statement. Theorem 2 shows that for any exponentially stable error dynamics on a compact set $z_s(t)$7, one may choose

$z_s(t)$8

and verify the RTF conditions by estimating the Lipschitz constant of the closed-loop error dynamics, the exponential decay rate $z_s(t)$9, the overshoot $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$0, and a suitable recurrence bound $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$1 (Liu et al., 1 Oct 2025).

The synthesis recipe is given in four steps:

1. Linearize or bound the error subsystem $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$2.
2. Choose a norm-based $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$3, or a quadratic form $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$4, and synthesize a low-level control gain such as PD or LQR to render the error exponentially stable.
3. Compute or over-approximate $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$5.
4. Form the RCBF $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$6 with $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$7 from Theorem 3 (Liu et al., 1 Oct 2025).

The paper explicitly states that no SOS or HJ reachability programs are needed: any norm of an exponentially stable error yields an RTF (Liu et al., 1 Oct 2025). In context, this positions the framework as a computational alternative when direct CBF construction for the FoM is intractable.

A concise summary of the synthesis ingredients is given below.

Component	Role	Specified form
RoM CBF $$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$8	Encodes reduced-order safety	$$e(t)=z(t)-z_s(t), \qquad \dot e(t)=\dot z(t)-\dot z_s(t).$$9
RTF $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$0	Certifies recurrent tracking contraction	$$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$1 or $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$2
RCBF $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$3	Lifts safety to FoM tracking coordinates	$$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$4

This synthesis structure suggests a modular workflow in which safety design and tracking design remain coupled through explicit constants rather than through a monolithic nonlinear certificate.

5. Layered control architecture and $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$5-recurrence

The framework is embedded in a layered control architecture. The high-level RoM computes a safe reference through a safety filter QP, while the low-level FoM controller tracks that reference (Liu et al., 1 Oct 2025).

More specifically, the paper describes the decomposition as follows. At the high level, the RoM solves a safety filter QP to compute a safe velocity $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$6 such that $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$7 keeps $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$8 in the safe set $$V:\mathbb{R}^n\times\mathbb{R}^n \to \mathbb{R}_{\ge 0}$$9. At the low level, the FoM controller has the form

$S\subset\mathbb{R}^n\times\mathbb{R}^n$0

which tracks $S\subset\mathbb{R}^n\times\mathbb{R}^n$1 (Liu et al., 1 Oct 2025).

Within this architecture, the RTF condition

$S\subset\mathbb{R}^n\times\mathbb{R}^n$2

guarantees three linked properties: within every $S\subset\mathbb{R}^n\times\mathbb{R}^n$3-window the error norm has shrunk exponentially; the composite function $S\subset\mathbb{R}^n\times\mathbb{R}^n$4 is control $S\subset\mathbb{R}^n\times\mathbb{R}^n$5-recurrent; and safety is maintained through Theorems 3 and 4 (Liu et al., 1 Oct 2025).

The paper’s interpretation is that $S\subset\mathbb{R}^n\times\mathbb{R}^n$6-recurrence allows transient tracking excursions but forces a re-entry into the budget set $S\subset\mathbb{R}^n\times\mathbb{R}^n$7 before error grows too large. This clarifies an important point of terminology: recurrence is not mere boundedness, and it is not equivalent to asymptotic decrease at each instant. It is a structured temporal constraint on when contraction must reoccur.

6. Illustrative example and failure modes

The paper’s case study is a 2D double integrator

$S\subset\mathbb{R}^n\times\mathbb{R}^n$8

with circular obstacles (Liu et al., 1 Oct 2025).

The construction uses the following components. The RoM is $S\subset\mathbb{R}^n\times\mathbb{R}^n$9 with

$0\in\mathrm{int}(S)$0

The high-level safe velocity $0\in\mathrm{int}(S)$1 is computed by a small QP enforcing

$0\in\mathrm{int}(S)$2

The low-level control is

$0\in\mathrm{int}(S)$3

which yields exponential error rate $0\in\mathrm{int}(S)$4. Then $0\in\mathrm{int}(S)$5 is an RTF, $0\in\mathrm{int}(S)$6 is computed, and the resulting recurrent barrier is

$0\in\mathrm{int}(S)$7

The simulation results reported in the paper show that when $0\in\mathrm{int}(S)$8 and the initial state satisfies $0\in\mathrm{int}(S)$9, the barrier $a_1,a_2,\beta,\tau>0$0 never dips below zero despite tracking transients. By contrast, if $a_1,a_2,\beta,\tau>0$1 or the initial $a_1,a_2,\beta,\tau>0$2, collisions occur, with $a_1,a_2,\beta,\tau>0$3 (Liu et al., 1 Oct 2025). Time-plots of $a_1,a_2,\beta,\tau>0$4 exhibit the permitted spikes followed by exponential decay, while $a_1,a_2,\beta,\tau>0$5 remains non-negative.

This example sharpens the interpretation of the theory. The condition $a_1,a_2,\beta,\tau>0$6 is not merely technical bookkeeping; within the paper’s construction it determines whether recurrent contraction of tracking error is sufficient to dominate the barrier decay rate inherited from the RoM CBF.

7. Significance, scope, and common misunderstandings

Within the paper’s stated scope, RTFs generalize classical CLFs by trading pointwise monotonic decay for a weaker finite-time recurrence condition that is systematically verifiable on compact sets (Liu et al., 1 Oct 2025). Their practical role is not to replace RoM CBFs, but to complement them: the RoM still carries the explicit safety certificate, while the RTF mediates the effect of FoM tracking error.

Several potential misunderstandings are clarified by the formulation itself. First, RTFs do not eliminate the need for a CBF; rather, they augment RoM CBFs to construct an RCBF for the FoM. Second, the guarantee is conditional on the initial state lying in

$a_1,a_2,\beta,\tau>0$7

and on satisfaction of the recurrence condition. Third, the framework does not claim monotone decay of the tracking measure; transient spikes are explicitly allowed. Fourth, the claimed computational advantage is specific: the paper states that no SOS or HJ reachability programs are needed for the RTF synthesis recipe, not that all safety-critical control computations disappear (Liu et al., 1 Oct 2025).

In broader methodological context, this suggests a reframing of Lyapunov-style tracking certification for layered safety systems. The essential certificate is no longer an everywhere-decreasing scalar function, but a function whose decrease is guaranteed recurrently over fixed horizons. The paper’s contribution is to show that this weaker temporal structure is still sufficient to lift reduced-order safety constraints to the full-order dynamics under the stated assumptions.