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Efficient space-time discretizations for tracking the boundaries of reachable sets

Janosch Rieger, Kyria Wawryk

TL;DR

The paper addresses the high computational burden of approximating reachable sets for nonlinear control systems by exploiting boundary tracking combined with nonuniform space-time discretization. It extends set-valued Euler methods to nonuniform grids through boundary-focused recursion and an abstract greedy subdivision scheme with provable finite-time convergence, backed by a concrete cost estimator and verification of key hypotheses. Numerically, the approach yields significant efficiency gains over uniform discretization and prior boundary methods, demonstrated via examples and runtime analyses. This framework enables scalable safety-oriented reachability analysis for complex nonlinear dynamics and opens avenues for semi-implicit extensions and stronger performance guarantees.

Abstract

The reachable sets of nonlinear control systems can in general only be numerically approximated, and are often very expensive to calculate. In this paper, we propose an algorithm that tracks only the boundaries of the reachable sets and that chooses the temporal and spatial discretizations in a non-uniform way to reduce the computational complexity.

Efficient space-time discretizations for tracking the boundaries of reachable sets

TL;DR

The paper addresses the high computational burden of approximating reachable sets for nonlinear control systems by exploiting boundary tracking combined with nonuniform space-time discretization. It extends set-valued Euler methods to nonuniform grids through boundary-focused recursion and an abstract greedy subdivision scheme with provable finite-time convergence, backed by a concrete cost estimator and verification of key hypotheses. Numerically, the approach yields significant efficiency gains over uniform discretization and prior boundary methods, demonstrated via examples and runtime analyses. This framework enables scalable safety-oriented reachability analysis for complex nonlinear dynamics and opens avenues for semi-implicit extensions and stronger performance guarantees.

Abstract

The reachable sets of nonlinear control systems can in general only be numerically approximated, and are often very expensive to calculate. In this paper, we propose an algorithm that tracks only the boundaries of the reachable sets and that chooses the temporal and spatial discretizations in a non-uniform way to reduce the computational complexity.

Paper Structure

This paper contains 16 sections, 18 theorems, 130 equations, 8 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Reachable sets and their approximations
  4. Reachable sets of differential inclusions
  5. The Euler scheme for uniform grids
  6. Boundary tracking for nonuniform grids
  7. Boundary pairs of digital images
  8. Boundary tracking for uniform grids
  9. An Euler scheme for nonuniform grids
  10. Boundary tracking for nonuniform grids
  11. An abstract subdivision scheme
  12. Efficient discretizations for boundary tracking
  13. Estimating the computational cost
  14. Verifying hypotheses P1 to P4
  15. Numerical examples
  16. ...and 1 more sections

Key Result

Lemma 3.2

For any numbers $\alpha,\rho>0$ with $\alpha\ge\rho/2$, the mapping is well-defined, and for all path-connected $M\in\mathcal{K}(\mathbb{R}^d)$, we have $\pi^\alpha_\rho(M)\in C_\rho$.

Figures (8)

  • Figure 1: Mathematically equivalent computations carried out either with entire sets (bottom row) or with their boundaries (top row).
  • Figure 2: Sets used in the calculation of $W_0^0,\ W_0^{-1}$ and $W_1$ in equations \ref{['eq:SDefns']} and \ref{['where:sigma:comes:from']}.
  • Figure 3: The dimension $d_R$ of the boundary of a set is determined by the dimension $d$ of the state space and the dimension $d_x$ of the set.
  • Figure 4: Approximate reachable sets of system \ref{['eqex:Simp7']} with $L=4$ generated by Algorithm \ref{['Alg:IterativeMethod']} with $3.7\%$ relative error. Resolution adjusted for printing purposes.
  • Figure 5: Discretization generated by Algorithm \ref{['Alg:IterativeMethod']} to approximate the reachable set of system \ref{['eqex:Simp7']} with $L=4$, and with $3.7\%$ relative error.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Definition 3.1
  • Lemma 3.2
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • Definition 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Definition 4.4
  • Corollary 4.5
  • ...and 31 more