# Using SOS and Sublevel Set Volume Minimization for Estimation of Forward Reachable Sets

## TL;DR

This paper introduces a convex Sum-of-Squares optimization method to compute tight outer approximations of forward reachable sets for nonlinear uncertain ODEs, minimizing volume via a determinant-like objective.

## Contribution

It presents a novel convex SOS-based approach for volume minimization in reachable set estimation, applicable to nonlinear systems with bounded disturbances.

## Key findings

- Effective approximation of Lorenz system's reachable set
- Accurate estimation for Van der Pol oscillator
- Demonstrates tight bounds with volume minimization

## Abstract

In this paper we propose a convex Sum-of-Squares optimization problem for finding outer approximations of forward reachable sets for nonlinear uncertain Ordinary Differential Equations (ODE's) with either (or both) L2 or point-wise bounded input disturbances. To make our approximations tight we seek to minimize the volume of our approximation set. Our approach to volume minimization is based on the use of a convex determinant-like objective function. We provide several numerical examples including the Lorenz system and the Van der Pol oscillator.