A Spectral Perspective on Stochastic Control Barrier Functions

arXiv:2603.19813 · eess.SY, cs.SY, math.OC · Submitted 2026-03-20 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: We're looking at the paper: "A Spectral Perspective on Stochastic Control Barrier Functions".

Dev: Here's the gist of the paper. Stochastic control barrier functions (SCBFs) provide a safety-critical control framework for systems subject to stochastic disturbances by bounding the probability of remaining within a safe set However, synthesizing a valid SCBF that explicitly reflects the true safety probability of the system, which is the most natural measure of safety, remains a challenge.

Rosa: The paper addresses how to characterize an SCBF that best captures the safety properties of a given stochastic system and how such an SCBF can be computed.

Dev: To answer this, the authors formalize a time-dependent Koopmanlike linear operator that governs the evolution of safety probability of the closed-loop system modeled as an Ito diffusion process.

Rosa: The dominant (biggest in absolute value) eigenvalue of this operator describes the global rate at which the closed-loop system’s safety probability decays over long time horizons, and therefore is an indicator for the system’s overall safety.

Title and authors: Rosa: So we're talking about this paper called "A Spectral Perspective on Stochastic Control Barrier Functions," and the authors are Inkyu Jang, Chams E. Mballo, Claire J. Tomlin, and H. Jin Kim > Would someone say that title out loud across a table?

Dev: It’s about taking a different mathematical angle to handle safety-critical control problems where there are random disturbances > Does that sound like something an engineer would say when describing a control system problem?

Taro: It suggests they're moving away from just finding any barrier function and instead looking at the system's underlying dynamics > That sounds like a step toward building something more robust than just plugging in numbers.

Rosa: Exactly, it frames the whole thing using this Koopman-like operator that describes how the safety probability of a system evolves over time > It’s about using linear operators to understand complex stochastic systems.

Dev: The core idea they are pushing is that the dominant eigenvalue of this operator gives us a direct measure of how fast safety probability drops over long periods > That eigenvalue is key to assessing overall system safety.

Taro: So, instead of just checking a fixed region for safety, they are using the dynamics to figure out what's safe over time > That shifts the focus from a static check to an evolving assessment.

The paper's summary: Rosa: They spend most of this paper formalizing how this operator governs the evolution of safety probability for a system that’s modeled as an Ito diffusion process > This gives them a solid mathematical foundation for what they are trying to do.

Dev: The main finding is that the dominant eigenfunction of this operator turns out to be a natural and valid choice for a control barrier function because it's directly linked to the safety probability itself > That means it’s inherently interpretable.

Taro: So, this eigenfunction isn't just some abstract math thing; it actually quantifies the relative long-term safety of any starting point in terms of where its trajectory is going > It tells you how safe a particular initial state is over a long time horizon compared to others.

Rosa: And they pair that with the dominant eigenvalue, which represents the largest one in absolute value, and that number tells you the global rate at which safety probability decays > That's a holistic measure for the entire closed-loop system.

Dev: They also propose a way to actually compute this whole eigenpair using something called power-policy iteration > It’s not just theory; they give you an algorithm to find both the SCBF and an optimized backup policy simultaneously.

Taro: That means instead of trying to guess the perfect safety function, you run this iteration and it finds the best one along with a control policy that actually works well > It makes finding a safe control strategy more systematic.

Rosa: So, what this means for us is we’re not just looking for *a* barrier function; we’re finding *the* natural one dictated by the system's dynamics > It’s about synthesis based on the system itself.

The paper's improvements: Taro: One of the main improvements they highlight is formalizing this operator theory framework, which allows them to characterize an SCBF that truly best captures a given stochastic system's safety properties > This moves beyond just guessing functions.

Dev: They are using this Koopman-like operator to describe the evolution of safety probability for systems modeled as Ito diffusion processes > That's the specific mathematical tool they rely on.

Rosa: The improvement lies in showing that this dominant eigenfunction is interpretable and meaningful because it’s explicitly connected to the safety probability, which makes it a natural choice for a barrier function > It’s not arbitrary.

Dev: And they propose power-policy iteration as a joint algorithm, which computes the SCBF and an optimized backup policy together > This is more efficient than doing them separately.

Taro: That iteration process is guaranteed to converge linearly with a positive real dominant eigenvalue having nonzero spectral gap, which shows it's a stable way to find those solutions > That guarantees convergence without getting stuck in weird behavior.

Rosa: So they are giving us a reliable method—this power-policy iteration—to synthesize the actual barrier functions we need in real applications > It’s a practical computational tool for safety engineers.

Conclusion: Dev: To wrap up this paper "A Spectral Perspective on Stochastic Control Barrier Functions," the main point is that the dominant eigenfunction is a natural and valid SCBF because it explicitly connects to the safety probability > This makes it interpretable for anyone using this framework.

Rosa: And that dominant eigenvalue gives us a holistic measure for system safety, showing exactly how fast that safety probability decays over long time horizons > It’s about understanding the long-term behavior of the closed-loop system.

Taro: For me, what this means is we get a way to quantify relative safety across different initial states using that eigenfunction, which is something we haven't done in this way before > That relative quantification across initial states is a big new capability here.

Dev: And the power-policy iteration procedure allows you to evolve an initially unsafe policy gradually into safer ones while simultaneously finding that optimal eigenpair > It’s about finding a safe path even when you start with a bad one.

Rosa: This spectral perspective on stochastic control barrier functions gives us a way to get interpretable safety metrics and a method to compute those functions through power-policy iteration > It’s a solid framework for getting natural safety metrics out of stochastic systems.

Seoul National University · University of California, Berkeley

eess.SY, cs.SY, math.OC

Submitted: 2026-03-20

Updated: 2026-10-28

Comments: 16 pages, 8 figures. This work has been submitted to the IEEE for possible publication

Code: https://github.com/HJReachability/helperOC

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 91/100

The gist: Stochastic control barrier functions (SCBFs) provide a safety-critical control framework for systems subject to stochastic disturbances by bounding the probability of remaining within a safe set

Key concepts

Stochastic Control Barrier Functions (SCBFs)
These are mathematical functions used to ensure a system stays within a safe region despite random disturbances. The paper aims to find an SCBF that naturally reflects the true safety probability, which is the most accurate measure of safety for such systems.
Koopman-like Linear Operator
This is a mathematical tool used to model how the probability of being in a safe state changes over time. It acts like a linear operator on functions describing system states, allowing researchers to analyze the evolution of safety probability using eigenvalue analysis.
Dominant Eigenfunction and Eigenvalue
The dominant eigenfunction shows how safe an initial state is over a long period compared to other states. The corresponding dominant eigenvalue represents the global rate at which the system's safety probability decays, giving a holistic measure of overall system safety.

Terminology

Summary

Stochastic control barrier functions (SCBFs) provide a safety-critical control framework for systems subject to stochastic disturbances by bounding the probability of remaining within a safe set However, synthesizing a valid SCBF that explicitly reflects the true safety probability of the system, which is the most natural measure of safety, remains a challenge.

Motivation for the Spectral Approach

The paper addresses how to characterize an SCBF that best captures the safety properties of a given stochastic system and how such an SCBF can be computed. To answer this, the authors formalize a time-dependent Koopmanlike linear operator that governs the evolution of safety probability of the closed-loop system modeled as an Ito diffusion process. The dominant (biggest in absolute value) eigenvalue of this operator describes the global rate at which the closed-loop system’s safety probability decays over long time horizons, and therefore is an indicator for the system’s overall safety. The corresponding eigenfunction, called the dominant eigenfunction, quantifies relative safety across initial states in terms of long-term safety probability of the closed-loop trajectories.

Formalization of Safety and SCBFs

The paper defines stochastic dynamics using an Ito stochastic differential equation (SDE). The safety probability is denoted by Zπ(t, x) = Pπ[X†t ≠ KX0 = x], which is a nonnegative and non-increasing function of t starting at the initial value Zπ(0, x) = 1C (x). The SCBF is introduced as a straightforward extension of Definition 1 using the infinitesimal generator of the stochastic process. Specifically, Definition 3 defines an SCBF h as a twice continuously differentiable function if there exists a Lipschitz continuous backup policy πb such that Aπb h(x) + γh(x) ≥ 0 for all x in Int C. Theorem 1 establishes the safety probability lower bound Zπ(t, x) ≥ h(x)/h0 · e−γt, where h0 = maxx∈C h(x).

Spectral Analysis of the Safety Operator

The safety probability is characterized by a linear operator Tπ acting on a space of scalar functions, which acts as a semigroup. Theorem 2 summarizes useful properties of the operator Tπt, stating that it is a non-expansive positive linear operator and that if t > 0, it is a compact operator. The paper shows that Tπt has a unique real dominant eigenvalue rπt ∈ (0, 1] which equals the spectral radius of Tπt, and a positive real dominant eigenfunction ψπt satisfying Tπt ψπt(x) = rπt ψπt(x). This dominant eigenpair is especially meaningful as the dominant eigenfunction ψπ quantitatively evaluates how safe a given initial point is over a long horizon relative to other initial states, and the dominant eigenvalue rπt provides a holistic safety measure for the overall closed-loop system through the lens of safety probability.

Computation via Power-Policy Iteration

A practical synthesis procedure called power-policy iteration is proposed to compute jointly the SCBF (i.e., the dominant eigenpair) and an optimized backup policy. The algorithm is extended by adding a policy update step so that it simultaneously computes the eigenpair and searches for a sufficiently safe backup policy π, as described in Algorithm 2. The iteration is guaranteed to converge linearly with positive real dominant eigenvalue with nonzero spectral gap. The power-policy iteration allows the initial policy that is potentially unsafe to gradually evolve into safer ones.

Empirical Validation and Results

The validity of this approach is demonstrated through simulations on safety-critical dynamics models. Numerical examples include a double integrator system with various disturbances, where the power-policy iteration converges to the same eigenvalue γπ = 1.2424 regardless of the initial guess. The converged eigenfunctions for remaining cases are visualized, and in some cases, they are not valid SCBFs in a strict sense but can still be used as an SCBF because escaping C through the discontinuity apparent in the results is impossible under the backup policy. The safety probability Zπ obtained from 1000 repeated trials is shown to be lower bounded by the bound provided by Theorem 1, and it is further verified that the asymptotic decay rate matches what is given by (39). The results for a WIG aircraft also show that the closed-loop trajectories stabilize to a safe steady flight under the safety-filtered policy.

Conclusion and Future Work

The paper presents an operator-theoretic framework for synthesizing valid SCBFs. By analyzing the Koopman-like operator governing the evolution of the system’s safety probability, it is demonstrated that its dominant eigenfunction constitutes a natural and valid SCBF for the system. The values of this dominant eigenfunction explicitly quantify the long-term safety probability of a given state, thereby offering a natural and interpretable safety metric. The corresponding eigenvalue, called the dominant eigenvalue, describes the global rate at which the safety probability decays and serves as an indicator for the overall safety of the closed-loop system. Future research directions include exploring non-dominant eigenmodes to provide tighter safety probability estimates over finite horizons and developing scalable approximation techniques, such as neural-network-based representations, for higher-dimensional systems. The authors thank several professors for insightful discussions.

Acknowledgment

The authors would like to thank Prof. Jason J. Choi (UCLA), Prof. Namhoon Cho (Seoul National University), Prof. Somil Bansal (Stanford University), Dylan Hirsch (UC San Diego), Jonghae Park (Seoul National University), and Sihyun Cho (Seoul National University) for insightful discussions.

Appendix A

Lemma 2 states that if X is a Banach space and L: X → X a compact linear operator, then its spectrum, σ(L), is a countable set with possible accumulation only at 0. Lemma 3 states that if Φ is a set of real-valued functions defined on a compact subspace of R n and is uniformly bounded and uniformly equicontinuous, then Φ is relatively compact. The Burkholder-Davis-Gundy Inequality provides bounds for the supremum of a local martingale.

Appendix B

The proof of Theorem 2 shows that Tπt is a non-expansive positive linear operator and that if t > 0, Tπt is a compact operator. The proof of Theorem 4 shows that the dominant eigenfunction is unique up to scalar multiplication, and the operator Tπt has nonzero spectral gap. The proof of Theorem 5 shows that Aπψ exists if and only if Aπψ exists, and Aπψ = Aπψ. The proof of Theorem 6 shows that Ttβ(x) = 0 for all x ∈ ∂C and t > 0.

Appendix C

The proof of Theorem 7 shows that the dominant eigenfunction remains unchanged with respect to time and the dominant eigenvalues also inherit the same semigroup structure because rπt+tψπ = Tπt+sψπ = rπt·Tπsψπ.

Improvements for AI systems

  1. Safety Critical Control Barrier Functions: The AI system can be equipped with a safety monitor that evaluates current barrier function values and their predicted rates of change under proposed inputs to ensure trajectories remain within a safe set, intervening by modifying the input to obtain a safe alternative.

  2. Spectral Safety Characterization: The system can determine the natural and valid SCBF by identifying the dominant eigenfunction of the Koopman-like linear operator, which explicitly quantifies relative long-term safety of the state.

  3. Global Safety Rate Estimation: The dominant eigenvalue can be used to indicate the global rate at which the safety probability decays, providing a holistic measure for the overall closed-loop system's safety.

  4. Power Policy Iteration Synthesis: An improved synthesis algorithm, power policy iteration, can be employed to jointly compute the dominant eigenpair and an optimized backup policy, leading to a safest backup policy among the set of admissible control policies.

  5. Optimized Control Synthesis: The system can synthesize a control input by solving an optimization problem constrained by safety conditions, specifically SCBF-QP, which returns a safe policy that minimally deviates from the reference in terms of the distance metric ||cdot||W.

Abstract

Stochastic control barrier functions (SCBFs) provide a safety-critical control framework for systems subject to stochastic disturbances by bounding the probability of remaining within a safe set. However, finding a valid SCBF that explicitly reflects the long-horizon behavior of the true safety probability remains a challenge. This paper addresses this issue by adopting a spectral perspective, utilizing the linear operator that governs the evolution of the closed-loop system's safety probability. We find that the dominant eigenpair of this Koopman-like operator encodes fundamental safety information of the stochastic system. The dominant eigenfunction is a natural and valid SCBF, with values that characterize relative asymptotic safety across initial states, while the dominant eigenvalue indicates the global asymptotic rate at which the safety probability decays. A practical algorithm for computation is proposed, termed power-policy iteration, which jointly estimates the dominant eigenpair while updating the backup policy with the aim of improving long-term safety. The method is validated using simulation experiments on safety-critical dynamics models.

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