arXiv subjects

William Sharpless

Publications and source records attributed to William Sharpless.

9 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.

eess.SY↗

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.

eess.SY↗

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.

eess.SY↗

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

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↗

arXiv2025

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.

cs.RO↗

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.

cs.AI↗

arXiv2025

Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions

Recent developments in autonomous driving and robotics underscore the necessity of safety-critical controllers. Control barrier functions (CBFs) are a popular method for appending safety guarantees to a general control framework, but they are notoriously difficult to generate beyond low dimensions. Existing methods often yield non-differentiable or inaccurate approximations that lack integrity, and thus fail to ensure safety. In this work, we use physics-informed neural networks (PINNs) to generate smooth approximations of CBFs by computing Hamilton-Jacobi (HJ) optimal control solutions. These reachability barrier networks (RBNs) avoid traditional dimensionality constraints and support the tuning of their conservativeness post-training through a parameterized discount term. To ensure robustness of the discounted solutions, we leverage conformal prediction methods to derive probabilistic safety guarantees for RBNs. We demonstrate that RBNs are highly accurate in low dimensions, and safer than the standard neural CBF approach in high dimensions. Namely, we showcase the RBNs in a 9D multi-vehicle collision avoidance problem where it empirically proves to be 5.5x safer and 1.9x less conservative than the neural CBFs, offering a promising method to synthesize CBFs for general nonlinear autonomous systems.

cs.RO↗

arXiv2024

Linear Supervision for Nonlinear, High-Dimensional Neural Control and Differential Games

As the dimension of a system increases, traditional methods for control and differential games rapidly become intractable, making the design of safe autonomous agents challenging in complex or team settings. Deep-learning approaches avoid discretization and yield numerous successes in robotics and autonomy, but at a higher dimensional limit, accuracy falls as sampling becomes less efficient. We propose using rapidly generated linear solutions to the partial differential equation (PDE) arising in the problem to accelerate and improve learned value functions for guidance in high-dimensional, nonlinear problems. We define two programs that combine supervision of the linear solution with a standard PDE loss. We demonstrate that these programs offer improvements in speed and accuracy in both a 50-D differential game problem and a 10-D quadrotor control problem.

math.OC↗