Recurrent Lyapunov Functions Overview

• Recurrent Lyapunov Functions (RLFs) are stability certificates that use finite-horizon recurrence to prove system stability, integrating control theory and dynamical systems.
• They relax the strict pointwise decay requirement by allowing temporary increases, provided an exponential decrease occurs within each recurrent window.
• Recent research extends RLFs into control applications by coupling them with barrier functions and data-driven policies to achieve practical stabilization and safety.

Recurrent Lyapunov Functions (RLFs) are Lyapunov-type certificates in which stability is encoded through recurrence rather than through pointwise monotone decay alone. In recent control formulations, an RLF requires that within every finite horizon there exists a return time at which an exponentially weighted Lyapunov quantity has decreased; in Conley-type dynamical systems, closely related objects are complete or strict Lyapunov functions whose neutral set coincides with a recurrent set such as the chain recurrent, strong chain recurrent, or generalized recurrent set (Liu et al., 1 Oct 2025, Siegelmann et al., 5 Oct 2025, Bernardi et al., 2017, Bernardi et al., 2019).

1. Terminology and scope

The acronym RLF is not uniform across the literature. In "Automatically Discovering Relaxed Lyapunov Functions for Polynomial Dynamical Systems" (Liu et al., 2011), RLF means Relaxed Lyapunov Function, defined through negativity of the first nonzero higher-order Lie derivative. In chemical-reaction-network work, Piecewise-Linear in Rates Lyapunov functions are described as robust with respect to arbitrary monotone kinetics, and are presented as very close to what later literature would call robust Lyapunov functions in reaction coordinates (Al-Radhawi et al., 2014). By contrast, the recurrence-based usage considered here is the one made explicit in recent control papers and strongly connected to Conley-style recurrence theory (Siegelmann et al., 5 Oct 2025, Liu et al., 1 Oct 2025).

In the recurrence-based sense, the defining relaxation is that Lyapunov decrease need not hold at every instant. Instead, decrease is required at recurrent times, or strict decrease is required only outside a distinguished recurrent set. This weaker requirement broadens the class of admissible certificates while preserving strong stability consequences in the settings where the associated theorems apply (Liu et al., 1 Oct 2025, Bernardi et al., 2017).

2. Finite-horizon recurrence as a Lyapunov condition

A recent explicit definition appears in "Safety-Critical Control via Recurrent Tracking Functions" (Liu et al., 1 Oct 2025). For an autonomous closed-loop trajectory $x(t)$, a function $V$ is an RLF over a compact $S\subseteq \mathcal X$ if it satisfies positive definiteness

$$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$

and exponential $\tau$-recurrence

$$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$

with $T_S(x):=\{t>0 \mid x(t)\in S\}$ and $T_S(x;\tau)=T_S(x)\cap (0,\tau]$ (Liu et al., 1 Oct 2025). The paper emphasizes that this is weaker than requiring $\dot V\le -\alpha V$ at every instant: $V$ may temporarily increase, and the error may transiently worsen, provided that within every bounded recurrence window there is a return event strong enough to compensate (Liu et al., 1 Oct 2025).

A control-theoretic extension is given in "Data-driven Practical Stabilization of Nonlinear Systems via Chain Policies" (Siegelmann et al., 5 Oct 2025). There, a Recurrent Control Lyapunov Function (R-CLF) over $V$0 is a continuous $V$1 satisfying

$V$2

together with control $V$3-exponential $V$4-recurrence

$V$5

This formulation yields practical exponential stabilizability rather than exact asymptotic convergence when $V$6 (Siegelmann et al., 5 Oct 2025).

A plausible synthesis is that modern recurrence-based RLFs are best understood as finite-horizon contraction certificates: the Lyapunov quantity is not required to decrease continuously, but it must decrease repeatedly after bounded waiting times.

3. Conley-type recurrence and complete Lyapunov functions

Long before the recent control definitions, recurrence-sensitive Lyapunov theory had been developed in Conley-style dynamical systems. In "Existence of complete Lyapunov functions with prescribed orbital derivative" (Giesl et al., 2021), the central object is the complete Lyapunov function for an autonomous ODE

$V$7

Such a function $V$8 is globally nonincreasing, strictly decreasing outside the chain-recurrent set $V$9, and constant on chain-transitive components in the precise sense that for every $S\subseteq \mathcal X$0, the set

$S\subseteq \mathcal X$1

is a chain transitive component. The paper’s main theorem states that for every compact $S\subseteq \mathcal X$2 and every negative $S\subseteq \mathcal X$3 function $S\subseteq \mathcal X$4 on a neighborhood of $S\subseteq \mathcal X$5, there exists a complete $S\subseteq \mathcal X$6-Lyapunov function $S\subseteq \mathcal X$7 such that

$S\subseteq \mathcal X$8

This provides a constructive realization theorem for recurrence-separating Lyapunov functions (Giesl et al., 2021).

For strong chain recurrence, "Existence of Lipschitz continuous Lyapunov functions strict outside the strong chain recurrent set" (Bernardi et al., 2017) proves that, for a continuous flow on a compact metric space that is uniformly Lipschitz continuous on compact subsets of time, there exists a Lipschitz continuous Lyapunov function $S\subseteq \mathcal X$9 with

$$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$0

where $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$1 is the neutral set. The same paper also characterizes the strong chain recurrent set as the intersection of the neutral sets of all Lipschitz continuous Lyapunov functions (Bernardi et al., 2017). "A Conley-type Lyapunov function for the strong chain recurrent set" (Bernardi et al., 2020) removes the additional uniform Lipschitz-in-time assumption and proves the existence of a continuous Lyapunov function strictly decreasing outside $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$2 for arbitrary continuous flows on compact metric spaces (Bernardi et al., 2020).

For the generalized recurrent set, "The generalized recurrent set, explosions and Lyapunov functions" (Bernardi et al., 2019) gives the exact characterization

$$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$3

where $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$4 is the set of continuous Lyapunov functions for the homeomorphism $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$5. It also states that there exists a continuous Lyapunov function $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$6 such that

$$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$7

and concludes that $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$8 admits a continuous strict Lyapunov function if and only if $$a_1 \|x-x^*\| \le V(x) \le a_2 \|x-x^*\|,\forall x \in S,$$9 (Bernardi et al., 2019).

Taken together, these results establish a broad recurrence-based Lyapunov paradigm: strict decrease organizes the nonrecurrent part of the dynamics, while neutrality or constancy identifies recurrent structure.

4. Control extensions: R-CLFs, recurrent tracking, and recurrent barriers

The control literature extends recurrence-based Lyapunov ideas from autonomous recurrence to closed-loop synthesis. In "Data-driven Practical Stabilization of Nonlinear Systems via Chain Policies" (Siegelmann et al., 5 Oct 2025), the R-CLF framework underlies Nonparametric Chain Policies (NCPs), where a finite library of verified control segments is applied sequentially. The paper proves that the norm candidate

$\tau$0

can serve as the recurrent certificate, and the abstract states an explicit sample complexity guarantee of

$\tau$1

number of trajectories. The policies are nonparametric, so new verified data can be readily incorporated to improve convergence rate or enlarge the certified region (Siegelmann et al., 5 Oct 2025).

"Safety-Critical Control via Recurrent Tracking Functions" (Liu et al., 1 Oct 2025) introduces Recurrent Tracking Functions (RTFs) as the tracking-side extension of RLFs. For tracking error variables $\tau$2 and $\tau$3, an RTF $\tau$4 over $\tau$5 satisfies

$\tau$6

and

$\tau$7

The paper proves that, under Lipschitz assumptions, an RTF implies exponential decay of the tracking-error rate,

$\tau$8

It then combines a reduced-order-model CBF $\tau$9 with the RTF through

$$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$0

obtaining a recurrent control barrier function (RCBF) whose zero-superlevel set is control $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$1-recurrent. When $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$2, the paper proves safety of the full-order model for all initial states in that recurrent set (Liu et al., 1 Oct 2025).

These control extensions preserve the central recurrence idea: bounded-horizon contractive events are sufficient for exponential convergence or safety, even when monotone decrease is unavailable or difficult to certify.

5. Sampled-data and learning-based relatives

A conceptually adjacent line of work replaces analytical recurrence inequalities by sampled one-step decrease conditions. "Off Policy Lyapunov Stability in Reinforcement Learning" (Gill et al., 11 Sep 2025) does not use the term RLF, but its Lyapunov mechanism is explicitly described as belonging to the same conceptual family as discrete-time or sampled-data Lyapunov methods that certify stability via repeated one-step decrease along closed-loop trajectories. Its key finite-difference condition is

$$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$3

with a margin-enforced training condition

$$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$4

The paper itself stresses that this is not a formal RLF paper, that certification is empirical and penalty-based, and that the proposed algorithms currently lack theoretical support (Gill et al., 11 Sep 2025).

"Neural Lyapunov Redesign" (Mehrjou et al., 2020) is also adjacent rather than definitional. It constructs an improving sequence of Lyapunov functions $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$5 tied to a sequence of feedback policies, alternating between ROA estimation and controller redesign. A plausible implication is that this is a form of algorithmic recurrence in Lyapunov certification: the Lyapunov object is repeatedly re-estimated after each controller update, but the paper does not define a recurrent Lyapunov function as a mathematical class (Mehrjou et al., 2020).

These learning-based variants indicate that recurrence-based Lyapunov reasoning is increasingly used in sampled, neural, and off-policy settings, but the strongest formal guarantees still belong to the explicit RLF, R-CLF, and Conley-type constructions rather than to the penalty-based approximations.

6. Limitations, ambiguities, and open issues

The principal technical limitation of recurrence-based Lyapunov conditions is that they are weaker than classical monotone decrease. In the RTF formulation, $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$6 may temporarily increase and the tracking error may transiently worsen; the guarantee is only that a sufficiently strong decrease occurs within each recurrence window (Liu et al., 1 Oct 2025). This weaker condition is often the source of flexibility, but it also changes the proof architecture: exponential convergence is recovered from repeated finite-horizon events plus Lipschitz continuity, not from a pointwise differential inequality (Siegelmann et al., 5 Oct 2025).

In dynamical-systems formulations, the recurrent set itself may depend sensitively on the chosen notion of recurrence. The strong chain recurrent set $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$7 is metric-dependent, unlike Conley’s chain recurrent set, and generalized recurrence can exhibit explosion phenomena under $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$8 perturbations even though chain recurrence does not (Bernardi et al., 2020, Bernardi et al., 2019). For homeomorphisms on compact manifolds of dimension at least $$\min_{t\in T_{S}(x;\tau)} e^{\alpha t} V(x(t)) \le V(x), \forall x \in S,$$9, the absence of such generalized-recurrence explosions is characterized by the existence of a decomposition of $T_S(x):=\{t>0 \mid x(t)\in S\}$0 without cycles (Bernardi et al., 2019).

Learning-based methods add further caveats. The off-policy RL formulation in (Gill et al., 11 Sep 2025) depends on sampled transitions, acknowledges off-policy bias, and explicitly states that the work shows promise in practice but currently lacks theoretical support. Even when the one-step decrease condition resembles an RLF-style contractive relation, the certificate remains dataset-dependent and empirical rather than universal (Gill et al., 11 Sep 2025).

Finally, the acronym itself remains a source of ambiguity. In polynomial stability analysis, RLF may denote a Relaxed Lyapunov Function (Liu et al., 2011); in reaction-coordinate stability analysis, later literature may use closely related robust-Lyapunov language for PWLR functions (Al-Radhawi et al., 2014). In the recurrence-based sense, however, the common thread is specific: an RLF is a Lyapunov object whose decrease is organized by recurrent times or whose neutral set identifies recurrent dynamics. That sense now spans Conley-style recurrence theory, finite-horizon control synthesis, and several emerging sampled-data generalizations (Siegelmann et al., 5 Oct 2025, Bernardi et al., 2017).