# Tropical Kraus maps for optimal control of switched systems

## TL;DR

This paper introduces tropical Kraus maps, a novel mathematical tool for approximating value functions in switched optimal control problems, offering improved scalability over existing methods.

## Contribution

The paper develops tropical analogues of Kraus maps and demonstrates their application in approximating value functions and solving switched control problems more efficiently.

## Key findings

- Major scalability improvements over previous schemes
- Effective approximation of value functions in switched control
- Successful application to joint spectral radius and Hamilton-Jacobi PDEs

## Abstract

Kraus maps (completely positive trace preserving maps) arise classically in quantum information, as they describe the evolution of noncommutative probability measures. We introduce tropical analogues of Kraus maps, obtained by replacing the addition of positive semidefinite matrices by a multivalued supremum with respect to the L\"owner order. We show that non-linear eigenvectors of tropical Kraus maps determine piecewise quadratic approximations of the value functions of switched optimal control problems. This leads to a new approximation method, which we illustrate by two applications: 1) approximating the joint spectral radius, 2) computing approximate solutions of Hamilton-Jacobi PDE arising from a class of switched linear quadratic problems studied previously by McEneaney. We report numerical experiments, indicating a major improvement in terms of scalability by comparison with earlier numerical schemes, owing to the "LMI-free" nature of our method.