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A New Framework for Bounding Reachability Probabilities of Continuous-time Stochastic Systems

Bai Xue

TL;DR

This work develops a barrier-certificate framework for bounding reachability probabilities in continuous-time stochastic systems modeled by SDEs, addressing both finite-horizon and exact-time bounds. By relaxing a parabolic PDE or applying Grönwall's inequality, the authors provide time-dependent barrier conditions that yield upper and lower bounds, offering alternatives to Doob-based methods. The approach is extended to reachability at specific time instants, using auxiliary stopped processes and associated generators, with a comprehensive SDP/SOS-based computational pipeline. Numerical examples demonstrate tighter bounds and practical applicability, and the work connects stochastic barrier methods to deterministic reach-avoid problems, outlining avenues for future numerical improvements and broader comparisons.

Abstract

This manuscript presents an innovative framework for constructing barrier functions to bound reachability probabilities for continuous-time stochastic systems described by stochastic differential equations (SDEs). The reachability probabilities considered in this paper encompass two aspects: the probability of reaching a set of specified states within a predefined finite time horizon, and the probability of reaching a set of specified states at a particular time instant. The barrier functions presented in this manuscript are developed either by relaxing a parabolic partial differential equation that characterizes the exact reachability probability or by applying the Grönwall's inequality. In comparison to the prevailing construction method, which relies on Doob's non-negative supermartingale inequality (or Ville's inequality), the proposed barrier functions provide stronger alternatives, complement existing methods, or fill gaps.

A New Framework for Bounding Reachability Probabilities of Continuous-time Stochastic Systems

TL;DR

This work develops a barrier-certificate framework for bounding reachability probabilities in continuous-time stochastic systems modeled by SDEs, addressing both finite-horizon and exact-time bounds. By relaxing a parabolic PDE or applying Grönwall's inequality, the authors provide time-dependent barrier conditions that yield upper and lower bounds, offering alternatives to Doob-based methods. The approach is extended to reachability at specific time instants, using auxiliary stopped processes and associated generators, with a comprehensive SDP/SOS-based computational pipeline. Numerical examples demonstrate tighter bounds and practical applicability, and the work connects stochastic barrier methods to deterministic reach-avoid problems, outlining avenues for future numerical improvements and broader comparisons.

Abstract

This manuscript presents an innovative framework for constructing barrier functions to bound reachability probabilities for continuous-time stochastic systems described by stochastic differential equations (SDEs). The reachability probabilities considered in this paper encompass two aspects: the probability of reaching a set of specified states within a predefined finite time horizon, and the probability of reaching a set of specified states at a particular time instant. The barrier functions presented in this manuscript are developed either by relaxing a parabolic partial differential equation that characterizes the exact reachability probability or by applying the Grönwall's inequality. In comparison to the prevailing construction method, which relies on Doob's non-negative supermartingale inequality (or Ville's inequality), the proposed barrier functions provide stronger alternatives, complement existing methods, or fill gaps.
Paper Structure (13 sections, 16 theorems, 81 equations, 4 figures, 4 tables)

This paper contains 13 sections, 16 theorems, 81 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounding Reachability Probabilities I
  4. Upper-bounding Reachability Probabilities
  5. Lower-bounding Reachability Probabilities
  6. Bounding Reachability Probabilities II
  7. Upper-bounding Reachability Probabilities
  8. Lower-bounding Reachability Probabilities
  9. Examples
  10. Conclusion
  11. Semi-definite Programming Implementation
  12. Connections to Deterministic Reach-avoid Problems
  13. Explanation on the Ineffectiveness of Condition \ref{['upper_bound_3II']}

Key Result

Proposition 1

Given system SDE, the infinitesimal generator $\mathcal{L}$ on a test function $v(t,\bm{x})$ is where $\frac{\partial v}{\partial t}$ and $\frac{\partial v}{\partial \bm{x}}$ represent the gradient of the test function $v(t,\bm{x})$ with respect to $t$ and $\bm{x}$, respectively, $\frac{\partial^2 v}{\partial \bm{x}^2}$ represents the second-order partial derivative of the test function $v(t,\

Figures (4)

  • Figure 1: Case 1 of Example 1. The region bounded by the red curve represents the safe set over the horizon $[0,100]$, the region bounded by the green curve represents the target set over the horizon $[0,100]$, and the blue curves represent five simulated trajectories starting from the initial state $x_0 = -0.8$.
  • Figure 2: Case 2 of Example 1. The region bounded by the red curve represents the safe set over the horizon $[0,1]$, the region bounded by the green curve represents the target set over the horizon $[0,1]$, and the blue curves represent five simulated trajectories starting from the initial state $x_0 = -0.8$.
  • Figure 3: Case 3 of Example 1. The region bounded by the red curve represents the safe set over the horizon $[0,10]$, the region bounded by the green curve represents the target set over the horizon $[0,10]$, and the blue curves represent five simulated trajectories starting from the initial state $x_0 = -0.5$.
  • Figure 4: Example 2. The region bounded by the red curve represents the safe set, the region bounded by the green curve represents the target set, and the blue curves represent five simulated trajectories starting from the initial state $(x_0,y_0)^{\top}=(-0.5,0.5)^{\top}$.

Theorems & Definitions (40)

  • Definition 1: oksendal2013stochastic
  • Proposition 1: oksendal2013stochastic
  • Definition 2: Reachability Probability I
  • Definition 3: Reachability Probability II
  • Lemma 1
  • proof
  • Lemma 2: Lemma 1, xue2023safe
  • Proposition 2
  • proof
  • Corollary 1
  • ...and 30 more