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Upper and Lower Bounds for a Class of Constrained Linear Time-Varying Games

Vincent Liu, Chris Manzie, Peter M. Dower

TL;DR

Addresses bounding the value function $v$ for constrained linear time-varying zero-sum differential games with a convex terminal cost $g$. The authors develop upper and lower bounds by constructing a viscosity supersolution $\bar{v}$ via trimmed-input trajectories and a viscosity subsolution $\underline{v}$ via hyperplanes, proving $\bar{v}\ge v\ge \underline{v}$ and achieving tightness on a characteristic set. An algorithmic procedure combines offline pre-computation of trajectory-costate data with online evaluation through a quadratic program or hyperplane maximization, yielding scalability not tied to state dimension. In reachability, these bounds translate to inner and outer approximations of backwards reachable sets, enabling rigorous safety analysis without grid-based solvers. The approach delivers substantial computational savings and rigorous guarantees for moderately high-dimensional systems, with tunable accuracy by the number of trajectories and hyperplanes.

Abstract

This paper develops an algorithm for upper- and lower-bounding the value function for a class of linear time-varying games subject to convex control sets. In particular, a two-player zero-sum differential game is considered where the respective players aim to minimise and maximise a convex terminal state cost. A collection of solutions of a single-player dynamical system subject to a trimmed control set is used to characterise a viscosity supersolution of a Hamilton-Jacobi (HJ) equation, which in turn yields an upper bound for the value function. Analogously, a collection of hyperplanes is used to characterise a viscosity subsolution of the HJ equation, which yields a lower bound. The computational complexity and memory requirement of the proposed algorithm scales with the number of solutions and hyperplanes that characterise the bounds, which is not explicitly tied to the number of system states. Thus, the algorithm is tractable for systems of moderately high dimension whilst preserving rigorous guarantees for optimal control and differential game applications.

Upper and Lower Bounds for a Class of Constrained Linear Time-Varying Games

TL;DR

Addresses bounding the value function for constrained linear time-varying zero-sum differential games with a convex terminal cost . The authors develop upper and lower bounds by constructing a viscosity supersolution via trimmed-input trajectories and a viscosity subsolution via hyperplanes, proving and achieving tightness on a characteristic set. An algorithmic procedure combines offline pre-computation of trajectory-costate data with online evaluation through a quadratic program or hyperplane maximization, yielding scalability not tied to state dimension. In reachability, these bounds translate to inner and outer approximations of backwards reachable sets, enabling rigorous safety analysis without grid-based solvers. The approach delivers substantial computational savings and rigorous guarantees for moderately high-dimensional systems, with tunable accuracy by the number of trajectories and hyperplanes.

Abstract

This paper develops an algorithm for upper- and lower-bounding the value function for a class of linear time-varying games subject to convex control sets. In particular, a two-player zero-sum differential game is considered where the respective players aim to minimise and maximise a convex terminal state cost. A collection of solutions of a single-player dynamical system subject to a trimmed control set is used to characterise a viscosity supersolution of a Hamilton-Jacobi (HJ) equation, which in turn yields an upper bound for the value function. Analogously, a collection of hyperplanes is used to characterise a viscosity subsolution of the HJ equation, which yields a lower bound. The computational complexity and memory requirement of the proposed algorithm scales with the number of solutions and hyperplanes that characterise the bounds, which is not explicitly tied to the number of system states. Thus, the algorithm is tractable for systems of moderately high dimension whilst preserving rigorous guarantees for optimal control and differential game applications.

Paper Structure

This paper contains 12 sections, 17 theorems, 86 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Viscosity Solutions
  4. Main Results
  5. Upper-bounding the Value Function
  6. Lower-bounding the Value Function
  7. Application: Reachability Analysis
  8. An Algorithmic Procedure
  9. Numerical Example
  10. Performance Evaluation
  11. Conclusion
  12. Supplementary Results

Key Result

Theorem 1

Consider the terminal value problem in which $g\in{\mathscr{C}}\left({\mathbb{R}}^n;\,{\mathbb{R}}\right)$ and the Hamiltonian $H$ is given by eq: prelim: hamiltonian. Then, eq: prelim: HJI with terminal condition admits the unique viscosity solution $v\in{\mathscr{C}}([0,T] \times{\mathbb{R}}^n;\,{\mathbb{R}})$ given by eq: prelim: v

Figures (5)

  • Figure 1: Illustration of the proposed upper bound $\Bar{g}$ in \ref{['eq: main: upper bound for g']}, which is the minimum over the individual upper bounds $\Bar{g}_k$ in \ref{['eq: main: upper bound for g via each level set']}. Points $\Bar{x}_{i,k}$, which characterise the upper bound, lie on the $\gamma_k$-level set of $g$.
  • Figure 2: Illustration of the proposed lower bound $\underline{g}$ in \ref{['eq: main: lower bound for g']}, which is the maximum over the individual lower bounds $\underline{g}_{i,k}$. The hyperplanes $\underline{g}_{i,k}$ support the epigraph of $g$ at $\Bar{x}_{i,k}$.
  • Figure 3: Slice along $x_2 = x_3 = 0$
  • Figure 4: Slice along $x_1 = x_3 = 0$
  • Figure 5: Slice along $x_1 = x_2 = 0$

Theorems & Definitions (39)

  • Definition 1
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Lemma 1
  • Lemma 2
  • proof
  • ...and 29 more