20,433 research outputs found
Computationally efficient approximations of the joint spectral radius
The joint spectral radius of a set of matrices is a measure of the maximal
asymptotic growth rate that can be obtained by forming long products of
matrices taken from the set. This quantity appears in a number of application
contexts but is notoriously difficult to compute and to approximate. We
introduce in this paper a procedure for approximating the joint spectral radius
of a finite set of matrices with arbitrary high accuracy. Our approximation
procedure is polynomial in the size of the matrices once the number of matrices
and the desired accuracy are fixed
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
Spontaneous pattern formation in an anti-ferromagnetic quantum gas
Spontaneous pattern formation is a phenomenon ubiquitous in nature, examples
ranging from Rayleigh-Benard convection to the emergence of complex organisms
from a single cell. In physical systems, pattern formation is generally
associated with the spontaneous breaking of translation symmetry and is closely
related to other symmetry-breaking phenomena, of which (anti-)ferromagnetism is
a prominent example. Indeed, magnetic pattern formation has been studied
extensively in both solid-state materials and classical liquids. Here, we
report on the spontaneous formation of wave-like magnetic patterns in a spinor
Bose-Einstein condensate, extending those studies into the domain of quantum
gases. We observe characteristic modes across a broad range of the magnetic
field acting as a control parameter. Our measurements link pattern formation in
these quantum systems to specific unstable modes obtainable from linear
stability analysis. These investigations open new prospects for controlled
studies of symmetry breaking and the appearance of structures in the quantum
domain
Extremal norms for positive linear inclusions
For finite-dimensional linear semigroups which leave a proper cone invariant
it is shown that irreducibility with respect to the cone implies the existence
of an extremal norm. In case the cone is simplicial a similar statement applies
to absolute norms. The semigroups under consideration may be generated by
discrete-time systems, continuous-time systems or continuous-time systems with
jumps. The existence of extremal norms is used to extend results on the
Lipschitz continuity of the joint spectral radius beyond the known case of
semigroups that are irreducible in the representation theory interpretation of
the word
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
On the Kinetics of Body versus End Evaporation and Addition of Supramolecular Polymers
Although pathway-specific kinetic theories are fundamentally important to
describe and understand reversible polymerisation kinetics, they come in
principle at a cost of having a large number of system-specific parameters.
Here, we construct a dynamical Landau theory to describe the kinetics of
activated linear supramolecular self-assembly, which drastically reduces the
number of parameters and still describes most of the interesting and generic
behavior of the system in hand. This phenomenological approach hinges on the
fact that if nucleated, the polymerisation transition resembles a phase
transition. We are able to describe hysteresis, overshooting, undershooting and
the existence of a lag time before polymerisation takes off, and pinpoint the
conditions required for observing these types of phenomenon in the assembly and
disassembly kinetics. We argue that the phenomenological kinetic parameter in
our theory is a pathway controller, i.e., it controls the relative weights of
the molecular pathways through which self-assembly takes place
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