Tropical Kraus maps for optimal control of switched systems

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.Comment: 15 page

A scalable algebraic method to infer quadratic invariants of switched systems

International audienceWe present a new numerical abstract domain based on ellip-soids designed for the formal verification of switched linearsystems. Unlike the existing approaches, this domain doesnot rely on a user-given template. We overcome the diffi-culty that ellipsoids do not have a lattice structure by ex-hibiting a canonical operator over-approximating the union.This operator is the only one which permits to perform anal-yses that are invariant with respect to a linear transforma-tion of state variables. Moreover, we show that this operatorcan be computed efficiently using basic algebraic operationson positive semidefinite matrices. We finally develop a fastnon-linear power-type algorithm, which allows one to de-termine sound quadratic invariants on switched systems ina tractable way, by solving fixed point problems over thespace of ellipsoids. We test our approach on several bench-marks, and compare it with the standard techniques basedon linear matrix inequalities, showing an important speedupon typical instances