On accuracy of approximation of the spectral radius by the Gelfand formula

The famous Gelfand formula $\rho(A)= \limsup_{n\to\infty}\|A^{n}\|^{1/n}$ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\rho(A)$. In the paper this deficiency is made up to some extent. By using the Bochi inequalities we establish explicit computable estimates for the rate of convergence of the quantities $\|A^{n}\|^{1/n}$ to $\rho(A)$. The obtained estimates are then extended for evaluation of the joint spectral radius of matrix sets.Comment: Corrected typos, improved estimate for the kay constant in the paper, added reference

A relaxation scheme for computation of the joint spectral radius of matrix sets

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.Comment: 16 pages, 2 figures, corrected typos, accepted for publication in JDE

An explicit Lipschitz constant for the joint spectral radius

In 2002 F. Wirth has proved that the joint spectral radius of irreducible compact sets of matrices is locally Lipschitz continuous as a function of the matrix set. In the paper, an explicit formula for the related Lipschitz constant is obtained.Comment: 9 pages, corrected typos and reference

A Sub-Neptune-sized Planet Transiting the M2.5 Dwarf G 9-40: Validation with the Habitable-zone Planet Finder

We validate the discovery of a 2-Earth-radii sub-Neptune-sized planet around the nearby high-proper-motion M2.5 dwarf G 9-40 (EPIC 212048748), using high-precision, near-infrared (NIR) radial velocity (RV) observations with the Habitable-zone Planet Finder (HPF), precision diffuser-assisted ground-based photometry with a custom narrowband photometric filter, and adaptive optics imaging. At a distance of d = 27.9 pc, G 9-40b is the second-closest transiting planet discovered by K2 to date. The planet's large transit depth (~3500 ppm), combined with the proximity and brightness of the host star at NIR wavelengths (J = 10, K = 9.2), makes G 9-40b one of the most favorable sub-Neptune-sized planets orbiting an M dwarf for transmission spectroscopy with James Webb Space Telescope, ARIEL, and the upcoming Extremely Large Telescopes. The star is relatively inactive with a rotation period of ~29 days determined from the K2 photometry. To estimate spectroscopic stellar parameters, we describe our implementation of an empirical spectral-matching algorithm using the high-resolution NIR HPF spectra. Using this algorithm, we obtain an effective temperature of T_(eff) = 3404±73K, and metallicity of [Fe/H] = −0.08±0.13. Our RVs, when coupled with the orbital parameters derived from the transit photometry, exclude planet masses above 11.7M⊕ with 99.7% confidence assuming a circular orbit. From its radius, we predict a mass of M = 5.0^(+3.8)_(−1.9) M⊕ and an RV semiamplitude of K = 4.1^(+3.1)_(−1.6) ms⁻¹, making its mass measurable with current RV facilities. We urge further RV follow-up observations to precisely measure its mass, to enable precise transmission spectroscopic measurements in the future

Iterative building of Barabanov norms and computation of the joint spectral radius for matrix sets

The problem of construction of Barabanov norms for analysis of properties of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. The method of Barabanov norms was the key instrument in disproving the Lagarias-Wang Finiteness Conjecture. The related constructions were essentially based on the study of the geometrical properties of the unit balls of some specific Barabanov norms. In this context the situation when one fails to find among current publications any detailed analysis of the geometrical properties of the unit balls of Barabanov norms looks a bit paradoxical. Partially this is explained by the fact that Barabanov norms are defined nonconstructively, by an implicit procedure. So, even in simplest cases it is very difficult to visualize the shape of their unit balls. The present work may be treated as the first step to make up this deficiency. In the paper two iteration procedure are considered that allow to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.Comment: 17 pages, 36 bibliography references, 3 figures; shortened version, new LaTeX style, fixed typos, accepted in DCDS-

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