Control Lyapunov Functions

Also known as CLF

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12 papers1 paper in the last four weeks, with none the four weeks before. 0.0% of all new papers.

Oct 13, 2026

A Rate Transfer Method: Convergence Rates in Stochastic Gradient Descent via Stochastic Calculus

We develop a rate-transfer method that converts finite-time Lyapunov estimates for an associated stochastic differential equation (SDE) into finite-time convergence bounds for stochastic gradient algorithms. Under generic moment and smoothness conditions, the convergence analysis of the stochastic gradient method reduces to a calculus argument verifying a Lyapunov generator inequality for the SDE. The method recovers established rates for classical stochastic gradient methods and provides a way to establish convergence rates for new optimization algorithms through calculus arguments applied to their associated SDEs.

Neil Walton

Durham University

Control Lyapunov FunctionsStochastic Optimization ConvergenceStochastic Gradient Descent
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Jul 19, 2026

Optimal Safety Control using High-Order Control Barrier Functions

This paper investigates the optimal safety control problem of nonlinear control systems by proposing novel high-order control barrier functions (HOCBFs). Different from zeroing HOCBFs, two novel HOCBFs are derived and the safety controllers are designed in an explicit way. Next, we implement vector Lyapunov function approach to propose a novel high-order control Lyapunov function (HOCLF) for the stabilization control problem. The relations between the proposed and existing HOCBFs are discussed. Afterwards, the compatibility of the proposed HOCLF and HOCBF is addressed to guarantee the stabilization and safety control objectives simultaneously, and thus the optimal controller is established. Finally, a numerical example from the navigation problem of quadrotors is presented to illustrate the efficacy of the derived results.

Neng Li, Zuodong Pan, Jiaxing Wang +2

Key Laboratory of Intelligent Control and Optimization for Industrial Equipment of Ministry of Education, Dalian University of Technology, Dalian 116024, China

Lyapunov-Based ControlSafety-Critical ControlControl Barrier FunctionsControl Lyapunov Functions
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Jun 26, 2026

Reachability Guarantees for Cart-Pole Swing-Up and Stabilization

The cart-pole swing-up is a canonical benchmark for nonlinear control of underactuated systems, yet an end-to-end guarantee linking the global swing-up maneuver to the local stabilizer is seldom formalized. We present a reachability analysis of a switched energy-based/LQR controller that certifies convergence to the upright equilibrium from a compact set of initial conditions. The swing-up law is derived from an energy-error Lyapunov function; canceling the autonomous conservative term yields a strictly sign-definite Lyapunov derivative, and convergence follows from LaSalle's invariance principle. We also propose an augmented Lyapunov function to regulate the steady-state cart velocity to zero, for which we establish almost-global convergence. For the controller handoff, a switching region is designed to lie strictly within the LQR region of attraction, formally certifying the swing-up-to-stabilization transition. Numerical simulations corroborate the theoretical analysis.

Mohamed Khalid M Jaffar

Department of Aerospace Engineering University of Maryland, College Park, USA

Reachability AnalysisLyapunov-Based ControlLyapunov StabilityControl Lyapunov Functions
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Jun 19, 2026

Conflict-Aware Switching for CBF-CLF-Based Multi-Goal Navigation

Quadratic programs (QPs) using Control Barrier Functions (CBFs) and Control Lyapunov Functions (CLFs) are widely used for safe control in reach-and-avoid navigation. However, the inherently conflicting nature of CBF and CLF constraints can lead to performance degradation, including slowdowns and deadlocks. This issue is exacerbated in multi-goal scenarios, where multiple nominal control objectives must be satisfied under shared safety constraints. Existing approaches for preemptive safety are often computationally expensive or overly conservative, while methods that relax or switch between nominal objectives are not well-suited for sequential goal-to-goal navigation. To address these limitations, we propose a conflict-aware switching strategy that detects high-conflict conditions and switches between available nominal control objectives to reduce constraint conflict. We apply this approach to multi-agent, multi-goal reach-and-avoid scenarios under CBF-CLF-QP control. Compared to a baseline sequential goal traversal strategy, our method reduces both completion time and timeout rates, demonstrating improved performance in satisfying all nominal control objectives while respecting safety constraints.

Rohan Walia, Kevin Leahy

Robotics Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Control Lyapunov FunctionsRobot NavigationControl Barrier FunctionsMulti-Agent ControlSafe Robot Navigation
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Jun 9, 2026

MARCH: Model-Assisted Reinforcement Learning for the Perceptive Control of Humanoids over Sparse Footholds

Perceptive bipedal locomotion over sparse terrain remains a difficult challenge: model-based methods are precise but brittle to uncertainty, while model-free methods are robust but struggle to discover the precise, constrained motions required for safety-critical locomotion where small errors can cause catastrophic failures. We propose a model-assisted reinforcement learning (RL) framework that combines both perspectives in three steps: (1) generate a safe reference trajectory using simplified models; (2) train a privileged teacher policy guided by a control Lyapunov function (CLF) reward built around the safe reference trajectory; and (3) distill the teacher into a vision-based student policy. We show that this model-assistance procedure produces physically grounded locomotion, improving sample efficiency, reducing the need for a complex learning curriculum, and achieving smoother locomotion behavior alongside stepping stone performance comparable to model-free baselines. We validate our approach in simulation and demonstrate successful deployment on a Unitree G1 humanoid robot navigating sparse footholds with lateral constraints.

Codrin Crismariu, Ryan K. Cosner

Department of Mechanical Engineering, Tufts University, Medford, MA, United States

Control Lyapunov FunctionsHumanoid Robot LocomotionTerrain-Aware Robot LocomotionRobot LocomotionRL for Legged LocomotionModel-Based RL
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May 29, 2026

Non-Asymptotic Convergence of Stochastic Iterative Algorithms: A Lyapunov Framework

We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations Fˉ(x)=x\bar{F}(x)=x, where the operator Fˉ(⋅)\bar{F}(\cdot) can only be accessed through a noisy oracle. We first focus on the standard setting in which Fˉ(⋅)\bar{F}(\cdot) is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.

Zaiwei Chen, Siva Theja Maguluri

Purdue IE · Georgia Tech IE

Control Lyapunov FunctionsLyapunov StabilityStochastic ApproximationTemporal-Difference LearningConvergence Analysis
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May 7, 2026

Learned Lyapunov Shielding for Adaptive Control

We augment the Slotine--Li adaptive controller for Euler--Lagrange systems with three learned components: a structured-quadratic Lyapunov function VψV_ψ whose positive-definiteness follows from a Cholesky parameterization, a residual Soft Actor--Critic policy that adds bounded torque corrections to the analytic baseline, and a physics-informed neural network that estimates unmodeled dynamics. A closed-form safety filter, derived from the single affine constraint V˙ψ+αVψ≤0\dot V_ψ+ αV_ψ\le 0, projects every policy output onto the safe set without requiring an online QP solver. We prove: global feasibility of the filter under a drift-decay condition on the control-degeneracy set; exponential stability under exact shielding, with a robust extension whose margin depends on the PINN approximation error; almost-sure convergence of the three-timescale policy--certificate--multiplier updates to a KKT point; and a PAC generalization bound for the certificate over compacts. On a 2-DOF manipulator with nonlinear friction and variable payload, the learned certificate accounts for most of the empirical gain: tracking error drops by 41% on nominal friction and 24% on aggressive friction at the centroid of the training distribution. A 7-DOF scalability study on a Franka Emika Panda confirms clean convergence of the full pipeline at industrial scale, identifies the conditions under which gains over exact model-based baselines should and should not be expected, and documents a warm-start pathology of the learned certificate that has practical implications for deployment.

Giansalvo Cirrincione, Adriano Fagiolini

Laboratoire LTI, Université de Picardie Jules Verne, Amiens, France. · MIRPALab, Department of Engineering, University of Palermo, 90128 Palermo, Italy.

Lyapunov-Based ControlSafe RLAdaptive ControlControl Lyapunov Functions
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May 3, 2026

Stability of Control Lyapunov Function Guided Reinforcement Learning

Reinforcement learning (RL) has become the de facto method for achieving locomotion on humanoid robots in practice, yet stability analysis of the corresponding control policies is lacking. Recent work has attempted to merge control theoretic ideas with reinforcement learning through control guided learning. A notable example of this is the use of a control Lyapunov function (CLF) to synthesize the reinforcement learning rewards, a technique known as CLF-RL, which has shown practical success. This paper investigates the stability properties of optimal controllers using CLF-RL with the goal of bridging experimentally observed stability with theoretical guarantees. The RL problem is viewed as an optimal control problem and exponential stability is proven in both continuous and discrete time using both core CLF reward terms and the additional terms used in practice. The theoretical bounds are numerically verified on systems such as the double integrator and cart-pole. Finally, the CLF guided rewards are implemented for a walking humanoid robot to generate stable periodic orbits.

Zachary Olkin, William D. Compton, Aaron D. Ames

The authors are with the Department of Control and Dynamical Systems at the California Institute of Technology.

RL ControlLyapunov StabilityRL for Legged LocomotionOptimal ControlRL StabilityControl Lyapunov Functions
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May 3, 2026

Zero-Shot, Safe and Time-Efficient UAV Navigation via Potential-Based Reward Shaping, Control Lyapunov and Barrier Functions

Autonomous navigation and obstacle avoidance remain a core challenge of modern Unmanned Aerial Vehicles (UAVs). While traditional control methods struggle with the complexity and variability of the environment, reinforcement learning (RL) enables UAVs to learn adaptive behaviors through interaction with the environment. Existing research with RL prioritizes the mission success at the expense of mission time and safety of UAVs. This study integrates Potential Based Reward Shaping (PBRS) with Control Lyapunov Functions (CLF) and Control Barrier Functions (CBF) to simultaneously optimize mission time and ensure formal safety guarantees. An RL model is trained in a generalized simple environment, then used in complex scenarios incorporating a CLF-CBF-QP filter without further training. Experimental results in simulated environments demonstrate a significant reduction in mission time and outstanding performance in complex environment.

Ashik Abrar Naeem, Mohammad Ariful Haque

RL ControlControl Lyapunov FunctionsUAV NavigationSafe Robot Navigation
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Mar 19, 2026

A Closed-Form CLF-CBF Controller for Whole-Body Continuum Soft Robot Collision Avoidance

Safe operation is essential for deploying robots in human-centered 3D environments. Soft continuum manipulators provide passive safety through mechanical compliance, but still require active control to achieve reliable collision avoidance. Existing approaches, such as sampling-based planning, are often computationally expensive and lack formal safety guarantees, which limits their use for real-time whole-body avoidance. This paper presents a closed-form Control Lyapunov Function--Control Barrier Function (CLF--CBF) controller for real-time 3D obstacle avoidance in soft continuum manipulators without online optimization. By analytically embedding safety constraints into the control input, the proposed method ensures stability and safety under the stated modeling assumptions, while avoiding feasibility issues commonly encountered in online optimization-based methods. The resulting controller is up to 10×10\times faster than standard CLF--CBF quadratic-programming approaches and up to 100×100\times faster than traditional sampling-based planners. Simulation and hardware experiments on a tendon-driven soft manipulator demonstrate accurate 3D trajectory tracking and robust obstacle avoidance in cluttered environments. These results show that the proposed framework provides a scalable and provably safe control strategy for soft robots operating in dynamic, safety-critical settings.

Kiwan Wong, Maximillian Stölzle, Wei Xiao +1

Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Robotics Engineering Department, Worcester Polytechnic Institute, Worcester, MA 01609, USA

Control Lyapunov FunctionsContinuum Robot ControlRobot Collision AvoidanceSoft RoboticsControl Barrier FunctionsLyapunov-Based Control
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Mar 19, 2026

Safety-Guaranteed Imitation Learning from Nonlinear Model Predictive Control for Spacecraft Close Proximity Operations

This paper presents a safety-guaranteed, runtime-efficient imitation learning framework for spacecraft close proximity control. We leverage Control Barrier Functions (CBFs) for safety certificates and Control Lyapunov Functions (CLFs) for stability as unified design principles across data generation, training, and deployment. First, a nonlinear Model Predictive Control (NMPC) expert enforces CBF constraints to provide safe reference trajectories. Second, we train a neural policy with a novel CBF-CLF-informed loss and DAgger-like rollouts with curriculum weighting, promoting data-efficiency and reducing future safety filter interventions. Third, at deployment a lightweight one-step CBF-CLF quadratic program minimally adjusts the learned control input to satisfy hard safety constraints while encouraging stability. We validate the approach for ESA-compliant close proximity operations, including fly-around with a spherical keep-out zone and final approach inside a conical approach corridor, using the Basilisk high-fidelity simulator with nonlinear dynamics and perturbations. Numerical experiments indicate stable convergence to decision points and strict adherence to safety under the filter, with task performance comparable to the NMPC expert while significantly reducing online computation. A runtime analysis demonstrates real-time feasibility on a commercial off-the-shelf processor, supporting onboard deployment for safety-critical on-orbit servicing.

Alexander Meinert, Niklas Baldauf, Peter Stadler +1

Space Applications Group, e:fs TechHub GmbH, Gaimersheim, Germany. · Faculty of Electrical Engineering and Information Technology, Ingolstadt University of Applied Sciences, Germany.

Control Lyapunov FunctionsImitation LearningSafety-Critical ControlSpacecraft Rendezvous and Proximity OperationsControl Barrier Functions
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Mar 11, 2026

Safety-critical Control Under Partial Observability: Reach-Avoid POMDP meets Belief Space Control

Partially Observable Markov Decision Processes (POMDPs) provide a principled framework for robot decision-making under uncertainty. Solving reach-avoid POMDPs, however, requires coordinating three distinct behaviors: goal reaching, safety, and active information gathering to reduce uncertainty. Existing online POMDP solvers attempt to address all three within a single belief tree search, but this unified approach struggles with the conflicting time scales inherent to these objectives. We propose a layered, certificate-based control architecture that operates directly in belief space, decoupling goal reaching, information gathering, and safety into modular components. We introduce Belief Control Lyapunov Functions (BCLFs) that formalize information gathering as a Lyapunov convergence problem in belief space, and show how they can be learned via reinforcement learning. For safety, we develop Belief Control Barrier Functions (BCBFs) that leverage conformal prediction to provide probabilistic safety guarantees over finite horizons. The resulting control synthesis reduces to lightweight quadratic programs solvable in real time, even for non-Gaussian belief representations with dimension >104>10^4. Experiments in simulation and on a space-robotics platform demonstrate real-time performance and improved safety and task success compared to state-of-the-art constrained POMDP solvers.

Matti Vahs, Joris Verhagen, Jana Tumova

Division of Robotics, Perception and Learning, KTH Royal Institute of Technology, Stockholm, Sweden · Digital Futures

Control Lyapunov FunctionsPartially Observable Markov Decision ProcessesBelief-Space PlanningControl Barrier FunctionsLyapunov-Based Control
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