# Reach-Avoid Problems with Time-Varying Dynamics, Targets and Constraints

arXiv: https://arxiv.org/abs/1410.6445 · HTML: https://mlsanity.com/paper/1410.6445/

## Key facts

- Cited by 44 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Oct 2014.

## Built on by (most cited first, 30 of 44)

- [The Safety Filter: A Unified View of Safety-Critical Control in Autonomous Systems](https://mlsanity.com/paper/2309.05837/) (2023)
- [Agile But Safe: Learning Collision-Free High-Speed Legged Locomotion](https://mlsanity.com/paper/2401.17583/) (2024)
- [Safety and Liveness Guarantees through Reach-Avoid Reinforcement Learning](https://mlsanity.com/paper/2112.12288/) (2021)
- [Generalizing Safety Beyond Collision-Avoidance via Latent-Space Reachability Analysis](https://mlsanity.com/paper/2502.00935/) (2025)
- [Iterative Reachability Estimation for Safe Reinforcement Learning](https://mlsanity.com/paper/2309.13528/) (2023)
- [ISAACS: Iterative Soft Adversarial Actor-Critic for Safety](https://mlsanity.com/paper/2212.03228/) (2022)
- [Sim-to-Lab-to-Real: Safe Reinforcement Learning with Shielding and Generalization Guarantees](https://mlsanity.com/paper/2201.08355/) (2022)
- [Safe Occlusion-aware Autonomous Driving via Game-Theoretic Active Perception](https://mlsanity.com/paper/2105.08169/) (2021)
- [Safe Model-Based Reinforcement Learning with an Uncertainty-Aware Reachability Certificate](https://mlsanity.com/paper/2210.07553/) (2022)
- [Gameplay Filters: Robust Zero-Shot Safety through Adversarial Imagination](https://mlsanity.com/paper/2405.00846/) (2024)
- [Formally Verifying Deep Reinforcement Learning Controllers with Lyapunov Barrier Certificates](https://mlsanity.com/paper/2405.14058/) (2024)
- [Hamilton-Jacobi Reachability in Reinforcement Learning: A Survey](https://mlsanity.com/paper/2407.09645/) (2024)
- [Scalable Learning of Safety Guarantees for Autonomous Systems using Hamilton-Jacobi Reachability](https://mlsanity.com/paper/2101.05916/) (2021)
- [Risk-averse autonomous systems: A brief history and recent developments from the perspective of optimal control](https://mlsanity.com/paper/2109.08947/) (2021)
- [Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertainty](https://mlsanity.com/paper/2210.05989/) (2022)
- [Deception Game: Closing the Safety-Learning Loop in Interactive Robot Autonomy](https://mlsanity.com/paper/2309.01267/) (2023)
- [Real-Time Out-of-Distribution Failure Prevention via Multi-Modal Reasoning](https://mlsanity.com/paper/2505.10547/) (2025)
- [Analyzing Human Models that Adapt Online](https://mlsanity.com/paper/2103.05746/) (2021)
- [To the Max: Reinventing Reward in Reinforcement Learning](https://mlsanity.com/paper/2402.01361/) (2024)
- [Human-AI Safety: A Descendant of Generative AI and Control Systems Safety](https://mlsanity.com/paper/2405.09794/) (2024)
- [Infinite-Horizon Reach-Avoid Zero-Sum Games via Deep Reinforcement Learning](https://mlsanity.com/paper/2203.10142/) (2022)
- [Correct-by-construction reach-avoid control of partially observable linear stochastic systems](https://mlsanity.com/paper/2103.02398/) (2021)
- [Verifiable Safety Q-Filters via Hamilton-Jacobi Reachability and Multiplicative Q-Networks](https://mlsanity.com/paper/2506.15693/) (2025)
- [Deep Reinforcement Learning-based UAV Navigation and Control: A Soft Actor-Critic with Hindsight Experience Replay Approach](https://mlsanity.com/paper/2106.01016/) (2021)
- [Safe and Reliable Training of Learning-Based Aerospace Controllers](https://mlsanity.com/paper/2407.07088/) (2024)
- [A Survey of Decision Making in Adversarial Games](https://mlsanity.com/paper/2207.07971/) (2022)
- [Pontryagin Neural Operator for Solving Parametric General-Sum Differential Games](https://mlsanity.com/paper/2401.01502/) (2024)
- [Your Learned Constraint is Secretly a Backward Reachable Tube](https://mlsanity.com/paper/2501.15618/) (2025)
- [Dual-Objective Reinforcement Learning with Novel Hamilton-Jacobi-Bellman Formulations](https://mlsanity.com/paper/2506.16016/) (2025)
- [Residual Neural Terminal Constraint for MPC-based Collision Avoidance in Dynamic Environments](https://mlsanity.com/paper/2508.03428/) (2025)

## Abstract (arXiv)

We consider a reach-avoid differential game, in which one of the players aims to steer the system into a target set without violating a set of state constraints, while the other player tries to prevent the first from succeeding; the system dynamics, target set, and state constraints may all be time-varying. The analysis of this problem plays an important role in collision avoidance, motion planning and aircraft control, among other applications. Previous methods for computing the guaranteed winning initial conditions and strategies for each player have either required augmenting the state vector to include time, or have been limited to problems with either no state constraints or entirely static targets, constraints and dynamics. To incorporate time-varying dynamics, targets and constraints without the need for state augmentation, we propose a modified Hamilton-Jacobi-Isaacs equation in the form of a double-obstacle variational inequality, and prove that the zero sublevel set of its viscosity solution characterizes the capture basin for the target under the state constraints. Through this formulation, our method can compute the capture basin and winning strategies for time-varying games at no additional computational cost with respect to the time-invariant case. We provide an implementation of this method based on well-known numerical schemes and show its convergence through a simple example; we include a second example in which our method substantially outperforms the state augmentation approach.