# Semidefinite Approximations of Reachable Sets for Discrete-time Polynomial Systems

## TL;DR

This paper introduces a method to compute certified outer approximations of the reachable set for discrete-time polynomial systems using a hierarchy of semidefinite relaxations, ensuring convergence to the true set.

## Contribution

It develops a novel hierarchy of semidefinite relaxations for approximating reachable sets, with convergence guarantees and practical numerical implementation.

## Key findings

- Hierarchies of super level sets approximate reachable sets with increasing accuracy.
- Convex semidefinite programs compute polynomial coefficients efficiently.
- Numerical examples demonstrate the method's effectiveness.

## Abstract

We consider the problem of approximating the reachable set of a discrete-time polynomial system from a semialgebraic set of initial conditions under general semialgebraic set constraints. Assuming inclusion in a given simple set like a box or an ellipsoid, we provide a method to compute certified outer approximations of the reachable set. The proposed method consists of building a hierarchy of relaxations for an infinite-dimensional moment problem. Under certain assumptions, the optimal value of this problem is the volume of the reachable set and the optimum solution is the restriction of the Lebesgue measure on this set. Then, one can outer approximate the reachable set as closely as desired with a hierarchy of super level sets of increasing degree polynomials. For each fixed degree, finding the coefficients of the polynomial boils down to computing the optimal solution of a convex semidefinite program. When the degree of the polynomial approximation tends to infinity, we provide strong convergence guarantees of the super level sets to the reachable set. We also present some application examples together with numerical results.