Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving Reach- and Stabilize-Avoid Problems Using Discounted Reachability

Boyang Li, Zheng Gong, Sylvia Herbert

TL;DR

This work addresses infinite-horizon reach-avoid and stabilize-avoid problems for general nonlinear continuous-time systems using Hamilton-Jacobi reachability and a novel discounted RA value function $V_\gamma$. It proves contraction of the Bellman backup and that $V_\gamma$ is the unique viscosity solution of a Hamilton-Jacobi-Isaacs VI, and shows exact recovery of the RA set as $\{x: V_\gamma(x) < 0\}$. For stabilization-avoid, it couples RA with Robust Control Lyapunov-Value Functions in a two-step framework to recover the SA set, and provides practical controller synthesis. The approach is validated on a 3D Dubins car, demonstrating reliable RA and SA performance under worst-case disturbances and illustrating the method's scalability and deterministic safety guarantees.

Abstract

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.

Solving Reach- and Stabilize-Avoid Problems Using Discounted Reachability

TL;DR

This work addresses infinite-horizon reach-avoid and stabilize-avoid problems for general nonlinear continuous-time systems using Hamilton-Jacobi reachability and a novel discounted RA value function . It proves contraction of the Bellman backup and that is the unique viscosity solution of a Hamilton-Jacobi-Isaacs VI, and shows exact recovery of the RA set as . For stabilization-avoid, it couples RA with Robust Control Lyapunov-Value Functions in a two-step framework to recover the SA set, and provides practical controller synthesis. The approach is validated on a 3D Dubins car, demonstrating reliable RA and SA performance under worst-case disturbances and illustrating the method's scalability and deterministic safety guarantees.

Abstract

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.
Paper Structure (13 sections, 9 theorems, 91 equations, 2 figures)

This paper contains 13 sections, 9 theorems, 91 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. BACKGROUND
  3. Dynamic System
  4. Reach-Avoid and Stabilize-Avoid Problems
  5. A NEW DISCOUNTED RA VALUE FUNCTION
  6. Definition and Properties
  7. Hamilton-Jacobi Characertization of $V_\gamma$
  8. Computation of $V_\gamma$
  9. The Finite-horizon Value Function and Control Synthesis
  10. TWO-STEP FORMULATION FOR STABILIZE-AVOID PROBLEM
  11. NUMERICAL EXAMPLES
  12. CONCLUSIONS
  13. Proof of the Uniqueness of the Viscosity Solution

Key Result

Theorem 1

The relative state can be exponentially stabilized to the $\mathcal{I}_{\text{m}}$ from $\mathcal{D}_\gamma \setminus \mathcal{I}_{\text{m}}$, if the R-CLVF exists in $\mathcal{D}_\gamma$, i.e., $\exists k > 0$, $\forall t \geq 0$

Figures (2)

  • Figure 1: SA value function $V_{\gamma}^{\text{SA}}(x)$ and its level sets. The regions enclosed by the green and black solid lines are the target $\mathcal{T}$ and obstacles, respectively, so the region outside the black solid lines is the constraint set $\mathcal{C}$. The regions enclosed by the magenta dashed lines indicate the zero superlevel set of $V_{\gamma}^{\text{SA}}(x)$, so the regions outside compose the $\mathcal{S} \mathcal{A}(\mathcal{T}, \mathcal{C})$ set.
  • Figure 2: SA (left) and RA (right) trajectories of the 3D Dubins car that starts at $[-4;4;0]$ and aims to reach the green disk (target set $\mathcal{T}$). Left: the car safely reaches $\mathcal{I_\text{M}}$ (blue dashed lines) using $\pi_{RA}$ from \ref{['eqn:RA_ctrl']} and then stabilizes to $\mathcal{I_{\text{m}}}$ (the torus region enclosed by red dashed lines) using $\pi_{H}$ from \ref{['eqn:SA_ctrl']}. Right: the car safely reaches $\mathcal{T}$ using $\pi_{RA}$ from \ref{['eqn:RA_ctrl']}, but does not remain in the target set. Instead, the trajectory will repeatedly leave and return.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • Definition 2
  • Proposition 1
  • Proposition 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 3
  • Corollary 1
  • ...and 2 more