arXiv subjects

Sylvia Herbert

Publications and source records attributed to Sylvia Herbert.

At least 19 recordsLinked to original sources

arXiv2026

Exact Decomposition of Value Functions for Two-Player Games in Hamilton-Jacobi Reachability

Hamilton-Jacobi reachability (HJR) is an important framework in safe control theory. HJR provides theoretical tools and numerical methods to obtain value-functions for tasks involving target-reaching and obstacle-avoidance. A recent work proposed an algebraic framework to scale to complex, composite tasks via decomposing the value functions of the composite tasks into value functions for the fundamental tasks traditionally studied in HJR, all in the one-player setting. Many of these decomposition results, however, do not directly translate to the two-player setting. In this technical note, we nevertheless show that one of the previous work's key results for analyzing various tasks involving reaching multiple targets while obeying constraints still holds in the two-player setting, so long as a critical monotonicity assumption is satisfied. In particular, we detail the analogous result and its proof (which structurally differs from the one-player case), we show via a counter-example what can go wrong if this monotonicity assumption is not satisfied, and we show how the analysis of a variety of tasks can be performed using this result. We note that, whereas the prior work considered discrete-time, infinite-horizon tasks, which are more standard in reinforcement learning, we here consider continuous-time, finite-horizon tasks, which are more standard in HJR.

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arXiv2026

MADR: MPC-guided Adversarial DeepReach

Hamilton-Jacobi (HJ) Reachability offers a framework for generating safe value functions and policies in the face of adversarial disturbance, but is limited by the curse of dimensionality. Physics-informed deep learning is able to overcome this infeasibility, but itself suffers from slow and inaccurate convergence, primarily due to weak PDE gradients and the complexity of self-supervised learning. A few works, recently, have demonstrated that enriching the self-supervision process with regular supervision (based on the nature of the optimal control problem), greatly accelerates convergence and solution quality, however, these have been limited to single player problems and simple games. In this work, we introduce MADR: MPC-guided Adversarial DeepReach, a general framework to robustly approximate the two-player, zero-sum differential game value function. In doing so, MADR yields the corresponding optimal strategies for both players in zero-sum games as well as safe policies for worst-case robustness. We test MADR on a multitude of high-dimensional simulated and real robotic agents with varying dynamics and games, finding that our approach significantly out-performs state-of-the-art baselines in simulation and produces impressive results in hardware.

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arXiv2026

Extending and Unifying the Fundamental Tasks of Hamilton-Jacobi Reachability Analysis

In this work, we introduce the generalized reach-avoid (GRA) task, which both extends and unifies the canonical tasks of Hamilton-Jacobi Reachability (HJR). We show that the GRA not only serves as a common primitive in this class of fundamental tasks, but also strictly extends the fundamental tasks that can be solved with HJR. Moreover, the GRA formulation enables one to compute the value functions of certain composite tasks, including ones from timed temporal logic, by decomposing the value function of the composite task into value functions of GRA tasks. We additionally show that the GRA is also a natural primitive to consider from a PDE perspective, as it can be used to represent all sufficiently regular solutions of the HJ-PDE that is canonical to HJR. Collectively, the results in this work show the theoretical and practical utility of this task within the increasingly important framework of HJR.

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arXiv2026

An Update to the Level Set Theorems in Hamilton-Jacobi Reachability Analysis

Hamilton-Jacobi Reachability (HJR) is an important framework for controlling safety-critical systems despite uncertainty. Its theoretical underpinnings are rooted in Hamilton-Jacobi Partial Differential Equations, which provide the value function used for controller synthesis. The Level Set Theorems of HJR allow one to interpret the value function in terms of satisfaction of a qualitative goal (e.g. goal-reaching or obstacle-avoidance). We here provide a technical update regarding additional criteria needed for these theorems to hold.

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arXiv2026

Exact Decomposition of Adversarial Dual-Objective Value Functions, with Applications to Optimal Drug Dosing

Hamilton-Jacobi Reachability (HJR) is a central framework in safe control theory. While HJR has traditionally focused on a few fundamental tasks, there is increasing interest in scaling to more complex objectives. Recent works have studied the exact decomposition of the value functions for two fundamental dual-objective tasks in the adversary-free setting. However, not all value function decompositions in HJR remain valid with an adversary. In this work, we develop theoretical approaches to certify that for these two composite value functions, the proposed decompositions still hold with an adversary. Finally, we show how these results can solve issues that arise when applying HJR to optimal drug regimen design.

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arXiv2026

Solving Reach- and Stabilize-Avoid Problems Using Discounted Reachability

In this article, we consider the infinite-horizon reach-avoid (RA) and stabilize-avoid (SA) zero-sum game problems for general nonlinear continuous-time systems, where the goal is to find the set of states that can be controlled to reach or stabilize to a target set, without violating constraints even under the worst-case disturbance. Based on the Hamilton-Jacobi reachability method, we address the RA problem by designing a new Lipschitz continuous RA value function, whose zero sublevel set exactly characterizes the RA set. We establish that the associated Bellman backup operator is contractive and that the RA value function is the unique viscosity solution of a Hamilton-Jacobi variational inequality. Finally, we develop a two-step framework for the SA problem by integrating our RA strategies with a recently proposed Robust Control Lyapunov-Value Function, thereby ensuring both target reachability and long-term stability. We numerically verify our RA and SA frameworks on a 3D Dubins car system to demonstrate the efficacy of the proposed approach.

math.OC↗

arXiv2026

Bellman Value Decomposition for Task Logic in Safe Optimal Control

Real-world tasks involve nuanced combinations of goal and safety specifications. In high dimensions, the challenge is exacerbated: formal automata become cumbersome, and the combination of sparse rewards tends to require laborious tuning. In this work, we consider the innate structure of the Bellman Value as a means to naturally organize the problem for improved automatic performance. Namely, we prove the Bellman Value for a complex task defined in temporal logic can be decomposed into a graph of Bellman Values, connected by a set of well-known Bellman equations (BEs): the Reach-Avoid BE, the Avoid BE, and a novel type, the Reach-Avoid-Loop BE. To solve the Value and optimal policy, we propose VDPPO, which embeds the decomposed Value graph into a two-layer neural net, bootstrapping the implicit dependencies. We conduct a variety of simulated and hardware experiments to test our method on complex, high-dimensional tasks involving heterogeneous teams and nonlinear dynamics. Ultimately, we find this approach greatly improves performance over existing baselines, balancing safety and liveness automatically.

cs.RO↗

arXiv2026

Value Functions for Temporal Logic: Optimal Policies and Safety Filters

While Bellman equations for basic reach, avoid, and reach-avoid problems are well studied, the relationship between value optimality and policy optimality becomes subtle in the undiscounted infinite-horizon setting, particularly for more complicated tasks. Greedily maximizing the Q-function can produce policies that indefinitely defer task completion for reach-avoid problems, or equivalently, Until specifications, even when the value function is optimal. Building upon recent results decomposing the value function for temporal logic (TL) into a graph of constituent value functions, we construct non-Markovian policies based on state history that avoid this pathology and prove their optimality with respect to the quantitative robustness score for nested Until, Globally, and Globally-Until specifications. We further show how the Q function can serve as a safety filter for complex TL specifications, extending prior results beyond simple avoid or reach-avoid tasks.

cs.RO↗

arXiv2026

Simultaneous improvement of control and estimation for battery management systems

Standard battery management systems treat the control and state estimation problems as decoupled objectives, relying on certainty equivalence controllers that are blind to the varying observability induced by nonlinear open-circuit voltage models. In this paper, we show that for a broad class of objectives, including the peak shaving and valley filling scenarios common in grid-connected energy storage, the expected cost of a stochastic battery system can be exactly parametrized by the conditional mean and covariance of the state of charge. This reformulation reveals a direct coupling between the control input and estimation quality, a coupling that certainty equivalence controllers ignore, and motivates a dual-control approach in which the controller actively reduces estimation uncertainty by driving the state to high observability regions without compromising the control objective. We derive a deterministic surrogate to this stochastic cost and pose the dual-control problem as a computationally tractable model predictive control problem. We validate our approach on a nine-battery system tracking a time-varying power/demand reference trajectory. We report simultaneous improvements in control cost (up to 20\% reduction) and state estimation error (up to 30\% reduction). The estimation improvement is reported across different state estimators: extended Kalman filter, unscented Kalman filter, and a moving horizon estimator, confirming that the estimation improvement of our approach is not restricted to a specific state observer.

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arXiv2026

Steering with Contingencies: Combinatorial Stabilization and Reach-Avoid Filters

In applications such as autonomous landing and navigation, it is often desirable to steer toward a target while retaining the ability to divert to at least $r$ (out of $p$) alternative sites if conditions change. In this work, we formalize this combinatorial contingency requirement and develop tractable control filters for enforcement. Combinatorial stabilization requires asymptotic stability of a selected equilibrium while ensuring the trajectory remains within the safe region of attraction of at least $r$-out-of-$p$ candidates. To enforce this requirement, we use control Lyapunov functions (CLFs) to construct regions of attraction, which are combined combinatorially within an optimization-based filter. Combinatorial targeting extends this framework to finite-horizon problems using Hamilton-Jacobi backward reach-avoid sets, accommodating shrinking reachable regions due to finite horizons or resource depletion. In both formulations, the resulting combinatorial stability filter and combinatorial reach-avoid filter require only $p+1$ constraints, preventing combinatorial blow-up and enabling safe real-time switching between targets. The framework is demonstrated on two examples where the filters ensure steering with contingency and enable safe diversion.

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arXiv2026

Robust Control Lyapunov-Value Functions for Nonlinear Disturbed Systems

Control Lyapunov Functions (CLFs) have been extensively used in the control community. A well-known drawback is the absence of a systematic way to construct CLFs for general nonlinear systems, and the problem can become more complex with input or state constraints. Our preliminary work on constructing Control Lyapunov Value Functions (CLVFs) using Hamilton-Jacobi (HJ) reachability analysis provides a method for finding a non-smooth CLF. In this paper, we extend our work on CLVFs to systems with bounded disturbance and define the Robust CLVF (R-CLVF). The R-CLVF naturally inherits all properties of the CLVF; i.e., it first identifies the "smallest robust control invariant set (SRCIS)" and stabilizes the system to it with a user-specified exponential rate. The region from which the exponential rate can be met is called the "region of exponential stabilizability (ROES)." We provide clearer definitions of the SRCIS and more rigorous proofs of several important theorems. Since the computation of the R-CLVF suffers from the "curse of dimensionality," we also provide two techniques (warmstart and system decomposition) that solve it, along with necessary proofs. Three numerical examples are provided, validating our definition of SRCIS, illustrating the trade-off between a faster decay rate and a smaller ROES, and demonstrating the efficiency of computation using warmstart and decomposition.

math.OC↗

arXiv2026

Refining Almost-Safe Value Functions on the Fly

Control Barrier Functions (CBFs) are a powerful tool for ensuring robotic safety, but designing or learning valid CBFs for complex systems is a significant challenge. While Hamilton-Jacobi Reachability provides a formal method for synthesizing safe value functions, it scales poorly and is typically performed offline, limiting its applicability in dynamic environments. This paper bridges the gap between offline synthesis and online adaptation. We introduce refineCBF for refining an approximate CBF - whether analytically derived, learned, or even unsafe - via warm-started HJ reachability. We then present its computationally efficient successor, HJ-Patch, which accelerates this process through localized updates. Both methods guarantee the recovery of a safe value function and can ensure monotonic safety improvements during adaptation. Our experiments validate our framework's primary contribution: in-the-loop, real-time adaptation, in simulation (with detailed value function analysis) and on physical hardware. Our experiments on ground vehicles and quadcopters show that our framework can successfully adapt to sudden environmental changes, such as new obstacles and unmodeled wind disturbances, providing a practical path toward deploying formally guaranteed safety in real-world settings.

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arXiv2026

Safe Stochastic Explorer: Enabling Safe Goal Driven Exploration in Stochastic Environments and Safe Interaction with Unknown Objects

Autonomous robots operating in unstructured, safety-critical environments, from planetary exploration to warehouses and homes, must learn to safely navigate and interact with their surroundings despite limited prior knowledge. Current methods for safe control, such as Hamilton-Jacobi Reachability and Control Barrier Functions, assume known system dynamics. Meanwhile existing safe exploration techniques often fail to account for the unavoidable stochasticity inherent when operating in unknown real world environments, such as an exploratory rover skidding over an unseen surface or a household robot pushing around unmapped objects in a pantry. To address this critical gap, we propose Safe Stochastic Explorer (S.S.Explorer) a novel framework for safe, goal-driven exploration under stochastic dynamics. Our approach strategically balances safety and information gathering to reduce uncertainty about safety in the unknown environment. We employ Gaussian Processes to learn the unknown safety function online, leveraging their predictive uncertainty to guide information-gathering actions and provide probabilistic bounds on safety violations. We first present our method for discrete state space environments and then introduce a scalable relaxation to effectively extend this approach to continuous state spaces. Finally we demonstrate how this framework can be naturally applied to ensure safe physical interaction with multiple unknown objects. Extensive validation in simulation and demonstrative hardware experiments showcase the efficacy of our method, representing a step forward toward enabling reliable widespread robot autonomy in complex, uncertain environments.

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arXiv2026

Learning to Nudge: A Scalable Barrier Function Framework for Safe Robot Interaction in Dense Clutter

Robots operating in everyday environments must navigate and manipulate within densely cluttered spaces, where physical contact with surrounding objects is unavoidable. Traditional safety frameworks treat contact as unsafe, restricting robots to collision avoidance and limiting their ability to function in dense, everyday settings. As the number of objects grows, model-based approaches for safe manipulation become computationally intractable; meanwhile, learned methods typically tie safety to the task at hand, making them hard to transfer to new tasks without retraining. In this work we introduce Dense Contact Barrier Functions(DCBF). Our approach bypasses the computational complexity of explicitly modeling multi-object dynamics by instead learning a composable, object-centric function that implicitly captures the safety constraints arising from physical interactions. Trained offline on interactions with a few objects, the learned DCBFcomposes across arbitrary object sets at runtime, producing a single global safety filter that scales linearly and transfers across tasks without retraining. We validate our approach through simulated experiments in dense clutter, demonstrating its ability to enable collision-free navigation and safe, contact-rich interaction in suitable settings.

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arXiv2025

Dual-Objective Reinforcement Learning with Novel Hamilton-Jacobi-Bellman Formulations

Hard constraints in reinforcement learning (RL) often degrade policy performance. Lagrangian methods offer a way to blend objectives with constraints, but require intricate reward engineering and parameter tuning. In this work, we extend recent advances that connect Hamilton-Jacobi (HJ) equations with RL to propose two novel value functions for dual-objective satisfaction. Namely, we address: 1) the Reach-Always-Avoid (RAA) problem -- of achieving distinct reward and penalty thresholds -- and 2) the Reach-Reach (RR) problem -- of achieving thresholds of two distinct rewards. In contrast with temporal logic approaches, which typically involve representing an automaton, we derive explicit, tractable Bellman forms in this context via decomposition. Specifically, we prove that the RAA and RR problems may be rewritten as compositions of previously studied HJ-RL problems. We leverage our analysis to propose a variation of Proximal Policy Optimization (DOHJ-PPO), and demonstrate that it produces distinct behaviors from previous approaches, outcompeting a number of baselines in success, safety and speed across a range of tasks for safe-arrival and multi-target achievement.

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arXiv2025

Viscosity CBFs: Bridging the Control Barrier Function and Hamilton-Jacobi Reachability Frameworks in Safe Control Theory

Control barrier functions (CBFs) and Hamilton-Jacobi reachability (HJR) are central frameworks in safe control. Traditionally, these frameworks have been viewed as distinct, with the former focusing on optimally safe controller design and the latter providing sufficient conditions for safety. A previous work introduced the notion of a control barrier value function (CB-VF), which is defined similarly to the other value functions studied in HJR but has certain CBF-like properties. In this work, we proceed the other direction by generalizing CBFs to non-differentiable ``viscosity'' CBFs. We show the deep connection between viscosity CBFs and CB-VFs, bridging the CBF and HJR frameworks. Through this bridge, we characterize the viscosity CBFs as precisely those functions which provide CBF-like safety guarantees (control invariance and smooth approach to the boundary). We then further show nice theoretical properties of viscosity CBFs, including their desirable closure under maximum and limit operations. In the process, we also extend CB-VFs to non-exponential anti-discounting and update the corresponding theory for CB-VFs along these lines.

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arXiv2025

Safe Event-triggered Gaussian Process Learning for Barrier-Constrained Control

While control barrier functions (CBFs) are employed in addressing safety, control synthesis methods based on them generally rely on accurate system dynamics. This is a critical limitation, since the dynamics of complex systems are often not fully known. Supervised machine learning techniques hold great promise for alleviating this weakness by inferring models from data. We propose a novel \revision{approach for safe event-triggered learning of Gaussian process models in CBF-based continuous-time control for unknown control-affine systems. By applying a finite excitation at triggering times, our approach ensures a sufficient information gain to maintain the feasibility of the CBF-based safety condition with high probability. Our approach probabilistically guarantees safety based on a suitable GP prior and rules out} Zeno behavior in the triggering scheme. The effectiveness of the proposed approach and theory is demonstrated in simulations.

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arXiv2025

Koopman-Hopf Hamilton-Jacobi Reachability and Control

The Hopf formula for Hamilton-Jacobi reachability (HJR) analysis has been proposed to solve high-dimensional differential games, producing the set of initial states and corresponding controller required to reach (or avoid) a target despite bounded disturbances. As a space-parallelizable method, the Hopf formula avoids the curse of dimensionality that afflicts standard dynamic-programming HJR, but is restricted to linear time-varying systems. To compute reachable sets for high-dimensional nonlinear systems, we pair the Hopf solution with Koopman theory for global linearization. By first lifting a nonlinear system to a linear space and then solving the Hopf formula, approximate reachable sets can be efficiently computed that are much more accurate than local linearizations. Furthermore, we construct a Koopman-Hopf disturbance-rejecting controller, and test its ability to drive a 10-dimensional nonlinear glycolysis model. We find that it significantly out-competes expectation-minimizing and game-theoretic model predictive controllers with the same Koopman linearization in the presence of bounded stochastic disturbance. In summary, we demonstrate a dimension-robust method to approximately solve HJR, allowing novel application to analyze and control high-dimensional, nonlinear systems with disturbance. An open-source toolbox in Julia is introduced for both Hopf and Koopman-Hopf reachability and control.

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