# Relaxing The Hamilton Jacobi Bellman Equation To Construct Inner And Outer Bounds On Reachable Sets

## TL;DR

This paper introduces a method to approximate both inner and outer bounds of reachable sets for polynomial systems by relaxing the Hamilton-Jacobi-Bellman equation, using polynomial bounds on the value function.

## Contribution

It presents a novel approach to construct provable bounds on reachable sets via polynomial sub- and super-value functions solving a relaxed HJB PDE.

## Key findings

- Polynomial bounds effectively approximate reachable sets.
- Inner and outer bounds have negligible Hausdorff distance at low polynomial degree.
- Method applicable to systems with polynomial dynamics and semialgebraic constraints.

## Abstract

We consider the problem of overbounding and underbounding both the backward and forward reachable set for a given polynomial vector field, nonlinear in both state and input, with a given semialgebriac set of initial conditions and with inputs constrained pointwise to lie in a semialgebraic set. Specifically, we represent the forward reachable set using the value function which gives the optimal cost to go of an optimal control problems and if smooth satisfies the Hamilton-Jacobi- Bellman PDE. We then show that there exist polynomial upper and lower bounds to this value function and furthermore, these polynomial sub-value and super-value functions provide provable upper and lower bounds to the forward reachable set. Finally, by minimizing the distance between these sub-value and super-value functions in the L1-norm, we are able to construct inner and outer bounds for the reachable set and show numerically on several examples that for relatively small degree, the Hausdorff distance between these bounds is negligible.